stochastic partial di erential equations: analysis and numerical...
TRANSCRIPT
Stochastic Partial Differential Equations:Analysis and Numerical Approximations
Arnulf Jentzen
September 14, 2015
2
Preface
These lecture notes have been written for the course “401-4606-00L Numerical Anal-ysis of Stochastic Partial Differential Equations” in the spring semester 2014 andin the spring semester 2015. These lecture notes are far away from being completeand remain under construction. In particular, these lecture notes do not yet con-tain a suitable comparison of the presented material with existing results, argumentsand notions in the literature. This will be the subject of a future version of theselecture notes. Furthermore, these lecture notes do not contain a number of proofs,arguments and intuitions. For most of this additional material, the reader is re-ferred to the lectures of the course “401-4606-00L Numerical Analysis of StochasticPartial Differential Equations” in the spring semester 2014. Sonja Cox and RyanKurniawan are gratefully acknowledged for their very helpful advice and assistance,especially for their help with the Matlab programs. Daniel Conus is also gratefullyacknowledged for several comments that helped to improve the presentation of theresults. In addition, we thank Antti Knowles for fruitful discussions on white noise.The students of the course “401-4606-00L Numerical Analysis of Stochastic PartialDifferential Equations” in the spring semester 2014 are gratefully acknowledged forpointing out a number of misprints to me. Special thanks are due to Timo Welti forbringing a number of misprints to my notice.
Zurich, February 2015
Arnulf Jentzen
3
Exercises
Solutions to the exercises can be turned in the designated mailbox in the anteroomHG G 53.x.
Exercise Exercises Deadlinesheet1 Exercises 1.1.8, 1.1.9, 2.2.6, and 2.4.4 05.03.2015, 10:15 AM2 Exercises 2.5.17, 2.5.20, 3.3.23, 3.5.2, and 3.5.3 19.03.2015, 10:15 AM3 Exercises 3.5.4, 3.5.15, 3.5.21, and 4.2.5 01.04.2015, 10:15 AM4 Exercises 4.3.4, 4.7.8, 5.3.22, and 6.1.6 15.04.2015, 10:15 AM5 Exercises 6.2.9, 6.2.11, 6.2.13, 6.2.14, and 6.2.21 24.04.2015, 10:15 AM6 Exercises 7.1.15, 8.1.14, and 8.1.15 07.05.2015, 10:15 AM7 Exercises 7.1.16, 7.2.2, and 8.2.5 14.05.2015, 10:15 AM8 Exercises 8.1.6, 9.1.6, and 9.4.1 21.05.2015, 10:15 AM
4
Contents
I Foundations in mathematical analysis 13
1 Gronwall-type inequalities 151.1 Properties of the beta and the gamma function . . . . . . . . . . . . 15
1.1.1 Functional equation of the gamma function . . . . . . . . . . . 161.1.2 Monotonicity properties of the gamma and the beta function . 161.1.3 Upper bounds for sums containing the beta and the gamma
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Integral operators related to the beta function . . . . . . . . . . . . . 191.3 Generalized exponential-type functions . . . . . . . . . . . . . . . . . 211.4 Generalized Gronwall-type inequalities . . . . . . . . . . . . . . . . . 21
1.4.1 Gronwall-type inequalities with a singularity at the initial time 241.4.2 Gronwall-type inequalities without a singularity at the initial
time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Regularity of nonlinear functions 272.1 General functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Nonlinear characterization of the Borel sigma-algebra . . . . . 282.2.2 Pointwise limits of measurable functions . . . . . . . . . . . . 292.2.3 Lp-sets of measurable functions for p P r0,8q . . . . . . . . . . 31
2.3 Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 Strongly measurable functions . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.2 Strongly measurable functions . . . . . . . . . . . . . . . . . . 332.4.3 Pointwise approximations of strongly measurable functions . . 342.4.4 Sums of strongly measurable functions . . . . . . . . . . . . . 362.4.5 Lp-spaces of strongly measurable functions for p P r0,8q . . . 37
2.5 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5
6 CONTENTS
2.5.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . 382.5.2 Semi-metric spaces . . . . . . . . . . . . . . . . . . . . . . . . 402.5.3 Continuity properties of functions . . . . . . . . . . . . . . . . 412.5.4 Modulus of continuity . . . . . . . . . . . . . . . . . . . . . . 422.5.5 Extensions of uniformly continuous functions . . . . . . . . . . 42
3 Linear functions 453.1 Linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 An intermezzo on sums over possibly uncountable index sets . . . . . 46
3.2.1 Fubini’s theorem in the case of non-sigma-finite measure spaces 463.2.2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.2.1 Confinal sequences . . . . . . . . . . . . . . . . . . . 473.2.3 Sums over possibly uncountable index sets . . . . . . . . . . . 493.2.4 Fubini for sums . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.1 Best approximations and projections in Hilbert spaces . . . . 533.3.2 Examples of orthonormal bases . . . . . . . . . . . . . . . . . 54
3.3.2.1 Trigonometric functions . . . . . . . . . . . . . . . . 543.3.2.2 Orthonormal basis in L2pBorelp0,1q; |¨|Rq . . . . . . . . 553.3.2.3 Transformations of orthonormal bases . . . . . . . . 61
3.4 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.1 Continuous linear functions on normed vector spaces . . . . . 633.4.2 Compact operators on Banach spaces . . . . . . . . . . . . . . 653.4.3 Nuclear operators on Banach spaces . . . . . . . . . . . . . . . 65
3.4.3.1 Definition of Nuclear operators . . . . . . . . . . . . 653.4.3.2 Relation of bounded linear operators and nuclear op-
erators . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4.3.3 Structure of the space of nuclear operators . . . . . . 683.4.3.4 Ideal property of the set of nuclear operators . . . . 693.4.3.5 Characterization of nuclear operators . . . . . . . . . 70
3.4.4 Hilbert-Schmidt operators on Hilbert spaces . . . . . . . . . . 703.4.4.1 Independence of the orthonormal basis . . . . . . . . 703.4.4.2 The Hilbert space of Hilbert-Schmidt operators . . . 713.4.4.3 Hilbert-Schmidt embeddings . . . . . . . . . . . . . . 72
3.5 Diagonal linear operators on Hilbert spaces . . . . . . . . . . . . . . . 733.5.1 Laplace operators on bounded domains . . . . . . . . . . . . . 74
3.5.1.1 Laplace operators with Dirichlet boundary conditions 753.5.1.2 Laplace operators with Neumann boundary conditions 76
CONTENTS 7
3.5.1.3 Laplace operators with periodic boundary conditions 773.5.2 Spectral decomposition for a diagonal linear operator . . . . . 783.5.3 Fractional powers of a diagonal linear operator . . . . . . . . . 803.5.4 Domain Hilbert space associated to a diagonal linear operator 813.5.5 Interpolation spaces associated to a diagonal linear operator . 82
3.6 The Bochner integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.6.1 Existence and uniqueness of the Bochner integral . . . . . . . 843.6.2 Definition of the Bochner integral . . . . . . . . . . . . . . . . 85
4 Semigroups of bounded linear operators 874.1 Definition of a semigroup of bounded linear operators . . . . . . . . . 874.2 Types of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.3 The generator of a semigroup . . . . . . . . . . . . . . . . . . . . . . 884.4 A global a priori bound for semigroups . . . . . . . . . . . . . . . . . 904.5 Strongly continuous semigroups . . . . . . . . . . . . . . . . . . . . . 90
4.5.1 A priori bounds for strongly continuous semigroups . . . . . . 904.5.2 Pointwise convergence in the space of bounded linear operators 924.5.3 Existence of solutions of linear ordinary differential equations
in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . 934.5.4 Domains of generators of strongly continuous semigroups . . . 944.5.5 Generators of strongly continuous semigroups . . . . . . . . . 954.5.6 A generalization of matrix exponentials to infinite dimensions 974.5.7 A characterization of strongly continuous semigroups . . . . . 98
4.6 Uniformly continuous semigroups . . . . . . . . . . . . . . . . . . . . 984.6.1 Matrix exponential in Banach spaces . . . . . . . . . . . . . . 994.6.2 Continuous invertibility of bounded linear operators in Banach
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.6.3 Generators of uniformly continuous semigroup . . . . . . . . . 1024.6.4 A characterization result for uniformly continuous semigroups 1034.6.5 An a priori bound for uniformly continuous semigroups . . . . 104
4.7 Semigroups generated by diagonal operators . . . . . . . . . . . . . . 1054.7.1 Semigroup generated by the Laplace operator . . . . . . . . . 1074.7.2 Smoothing effect of the semigroup . . . . . . . . . . . . . . . . 108
II Foundations in probability theory 111
5 Random variables with values in infinite dimensional spaces 1135.1 Borel sigma-algebras on normed vector spaces . . . . . . . . . . . . . 113
8 CONTENTS
5.1.1 The Hahn-Banach theorem . . . . . . . . . . . . . . . . . . . . 1135.1.2 Norm representations in normed vector spaces . . . . . . . . . 1145.1.3 Linear characterization of the Borel sigma-algebra . . . . . . . 115
5.2 Measures on normed vector spaces . . . . . . . . . . . . . . . . . . . . 1165.2.1 Uniqueness theorem for measures . . . . . . . . . . . . . . . . 1165.2.2 Fourier transform of a measure . . . . . . . . . . . . . . . . . 117
5.2.2.1 Characteristic functionals . . . . . . . . . . . . . . . 1175.2.2.2 Fourier transform on separable normed vector spaces 1185.2.2.3 Almost surely separably supported . . . . . . . . . . 1195.2.2.4 Trace set . . . . . . . . . . . . . . . . . . . . . . . . 1205.2.2.5 Fourier transform on normed vector spaces . . . . . . 123
5.2.3 Covariance of a measure . . . . . . . . . . . . . . . . . . . . . 1255.2.3.1 The Baire category theorem on complete metric spaces1255.2.3.2 Regularities for correlations on normed vector spaces 1255.2.3.3 Covariances of measures and random variables . . . . 128
5.2.4 Gaussian measures on normed vector spaces . . . . . . . . . . 1295.2.4.1 Fourier transform of a Gaussian measure . . . . . . . 131
5.3 Probability measures on Hilbert spaces . . . . . . . . . . . . . . . . . 1325.3.1 Nuclear operators on Hilbert spaces . . . . . . . . . . . . . . . 1325.3.2 Expectation and covariance operator . . . . . . . . . . . . . . 1365.3.3 Karhunen-Loeve expansion . . . . . . . . . . . . . . . . . . . . 1385.3.4 Gaussian measures on Hilbert spaces . . . . . . . . . . . . . . 139
5.3.4.1 Karhunen-Loeve expansion . . . . . . . . . . . . . . 1395.3.4.2 Construction of Gaussian measures on Hilbert spaces 1405.3.4.3 Karhunen-Loeve expansion for Brownian motion . . 144
6 Stochastic processes 1536.1 Hilbert space valued stochastic processes . . . . . . . . . . . . . . . . 153
6.1.1 Standard Wiener processes . . . . . . . . . . . . . . . . . . . . 1536.1.2 Pseudo inverse . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.2.1 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.2.2 Lenglart’s inequality . . . . . . . . . . . . . . . . . . . . . . . 1586.2.3 Modifications and indistinguishability . . . . . . . . . . . . . . 1626.2.4 Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . 1646.2.5 Construction of the stochastic integral . . . . . . . . . . . . . 1656.2.6 Elementary processes revisited . . . . . . . . . . . . . . . . . . 1746.2.7 Cylindrical Wiener process . . . . . . . . . . . . . . . . . . . . 176
CONTENTS 9
III Stochastic Partial Differential Equations (SPDEs) 179
7 Solutions of SPDEs 1817.1 Existence, uniqueness and properties of mild solutions of SPDEs . . . 181
7.1.1 Mild solutions of SPDEs . . . . . . . . . . . . . . . . . . . . . 1817.1.2 A setting for SPDEs with globally Lipschitz continuous non-
linearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837.1.3 A strong perturbation estimate for SPDEs . . . . . . . . . . . 1847.1.4 Uniqueness of mild solutions of SPDEs . . . . . . . . . . . . . 188
7.1.4.1 Uniqueness of predictable mild solutions of SEEs withglobally Lipschitz continuous coefficients . . . . . . . 188
7.1.4.2 Uniqueness of left-continuous mild solutions of SEEswith semi-globally Lipschitz continuous coefficients . 188
7.1.5 Existence and regularity of mild solutions of SPDEs . . . . . . 1927.1.6 A priori bounds for mild solutions of SPDEs . . . . . . . . . . 1927.1.7 Temporal-regularity of solution processes of SPDEs . . . . . . 1977.1.8 Existence of continuous solutions . . . . . . . . . . . . . . . . 198
7.2 Examples of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.2.1 Second order SPDEs . . . . . . . . . . . . . . . . . . . . . . . 200
IV Numerical Analysis of SPDEs 205
8 Strong numerical approximations for SPDEs 2078.1 Spatial spectral Galerkin approximations for SPDEs . . . . . . . . . . 207
8.1.1 Galerkin projections . . . . . . . . . . . . . . . . . . . . . . . 2078.1.2 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2148.1.3 A strong numerical approximation result for spectral Galerkin
approximations of SPDEs . . . . . . . . . . . . . . . . . . . . 2148.2 Temporal numerical approximations for SPDEs . . . . . . . . . . . . 218
8.2.1 Euler type approximations for SPDEs . . . . . . . . . . . . . . 2198.2.1.1 Exponential Euler method . . . . . . . . . . . . . . . 2198.2.1.2 Accelerated exponential Euler method . . . . . . . . 2208.2.1.3 Linear-implicit Euler method . . . . . . . . . . . . . 2218.2.1.4 Linear-implicit Crank-Nicolson-Euler method . . . . 222
8.2.2 Nonlinearity-stopped Euler type approximations for SPDEs . . 2238.2.2.1 Nonlinearity-stopped exponential Euler method . . . 2248.2.2.2 Nonlinearity-stopped linear-implicit Euler method . . 225
8.2.3 Milstein type approximations for SPDEs . . . . . . . . . . . . 226
10 CONTENTS
8.2.3.1 Exponential Milstein method . . . . . . . . . . . . . 2268.2.3.2 Linear-implicit Milstein method . . . . . . . . . . . . 2288.2.3.3 Linear-implicit Crank-Nicolson-Milstein method . . . 229
8.2.4 Strong convergence analysis for exponential Euler approxima-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.3 Noise approximations for SPDEs . . . . . . . . . . . . . . . . . . . . 2418.3.1 Noise perturbation estimates . . . . . . . . . . . . . . . . . . . 2418.3.2 Noise approximations for SPDEs . . . . . . . . . . . . . . . . 242
8.4 Full discretizations for SPDEs . . . . . . . . . . . . . . . . . . . . . . 2458.4.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2458.4.2 Full-discrete spectral Galerkin exponential Euler method for
SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2458.4.3 Full-discrete spectral Galerkin linear-implicit Euler method for
SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2488.4.4 Full-discrete spectral Galerkin nonlinearity-stopped exponen-
tial Euler method for SPDEs . . . . . . . . . . . . . . . . . . . 2508.4.5 Full-discrete spectral Galerkin nonlinearity-stopped linear-implicit
Euler method for SPDEs . . . . . . . . . . . . . . . . . . . . . 251
9 Weak numerical approximations for SPDEs 2559.1 An Ito type formula for SPDEs . . . . . . . . . . . . . . . . . . . . . 255
9.1.1 A setting for mild stochastic calculus . . . . . . . . . . . . . . 2559.1.2 Mild stochastic processes . . . . . . . . . . . . . . . . . . . . . 2569.1.3 Mild Ito formula . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.2 Solution processes of SPDEs . . . . . . . . . . . . . . . . . . . . . . . 2609.3 Transformations of semigroups of solutions of SPDEs . . . . . . . . . 2619.4 Weak convergence for temporal numerical approximations for SPDEs 2639.5 Weak convergence of Galerkin projections for SPDEs . . . . . . . . . 263
9.5.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2639.5.2 Weak convergence for spatial spectral Galerkin projections . . 264
10 Additional material 26910.1 Egorov’s theorem on almost uniform convergence . . . . . . . . . . . 269
10.1.1 General measure spaces . . . . . . . . . . . . . . . . . . . . . 26910.1.1.1 Almost sure convergence . . . . . . . . . . . . . . . . 26910.1.1.2 Luzin uniform-type convergence . . . . . . . . . . . . 27110.1.1.3 Almost uniform convergence . . . . . . . . . . . . . . 271
10.1.2 Finite measure spaces . . . . . . . . . . . . . . . . . . . . . . 272
CONTENTS 11
10.1.3 Sigma-finite measure spaces . . . . . . . . . . . . . . . . . . . 27310.1.4 General measure spaces . . . . . . . . . . . . . . . . . . . . . 274
10.2 Fast convergence in probability . . . . . . . . . . . . . . . . . . . . . 27410.3 Dini’s theorem on pointwise convergence of continuous functions . . . 275
10.3.1 On the compactness of the argument space . . . . . . . . . . . 27710.3.2 On the monotonicity of the approximating functions . . . . . . 27710.3.3 On the continuity of the approximating functions . . . . . . . 27810.3.4 On the continuity of the limit function . . . . . . . . . . . . . 279
11 Solutions to selected exercises 28111.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
11.1.1 Solution to Exercise 2.2.6 . . . . . . . . . . . . . . . . . . . . 28111.1.2 Solution to Exercise 2.4.4 . . . . . . . . . . . . . . . . . . . . 281
11.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28211.2.1 Solution to Exercise 3.5.2 . . . . . . . . . . . . . . . . . . . . 282
11.3 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28311.3.1 Solution to Exercise 7.1.15 . . . . . . . . . . . . . . . . . . . . 28311.3.2 Solution to Exercise 7.1.16 . . . . . . . . . . . . . . . . . . . . 28511.3.3 Solution to Exercise 7.2.2 . . . . . . . . . . . . . . . . . . . . 286
11.4 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28611.4.1 Solution to Exercise 8.1.6 . . . . . . . . . . . . . . . . . . . . 28611.4.2 Solution to Exercise 8.1.15 . . . . . . . . . . . . . . . . . . . . 28811.4.3 Solution to Exercise 8.2.5 . . . . . . . . . . . . . . . . . . . . 291
11.5 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29211.5.1 Solution to Exercise 9.4.1 . . . . . . . . . . . . . . . . . . . . 292
12 CONTENTS
Part I
Foundations inmathematical analysis
13
Chapter 1
Gronwall-type inequalities
This chapter is based on Section 7.1 in Henry [10].
1.1 Properties of the beta and the gamma func-
tion
For completeness we first recall the definition of the beta function and the gammafunction.
Definition 1.1.1 (Beta function and gamma function). We denote by B : p0,8q2 Ñp0,8q and Γ: p0,8q Ñ p0,8q the functions with the property that for all x, y P p0,8qit holds that
Bpx, yq “
ż 1
0
tpx´1qp1´ tqpy´1q dt (1.1)
and
Γpxq “
ż 8
0
tpx´1q e´t dt, (1.2)
we call B the beta function, and we call Γ the gamma function.
15
16 CHAPTER 1. GRONWALL-TYPE INEQUALITIES
1.1.1 Functional equation of the gamma function
Lemma 1.1.2 (Basic properties of the gamma function and the Beta function). Forall x, y P p0,8q, n P N0 it holds that
Bpx, yq “ Bpy, xq “Γpxq ¨ Γpyq
Γpx` yq“
ż 8
0
tpx´1q
p1` tqpx`yqdt, (1.3)
Γpn` 1q “ n! and Γpx` 1q “ x ¨ Γpxq . (1.4)
Proof of Lemma 1.1.2. First, observe that the integral transformation theorem en-sures that for all x, y P p0,8q it holds that
Bpx, yq “
ż 1
0
tpx´1qp1´ tqpy´1q dt “
ż 8
1
“
1t
‰px´1q “1´ 1
t
‰py´1q 1t2dt
“
ż 8
1
tp´x´1q“
t´1t
‰py´1qdt “
ż 8
1
tp´x´yq pt´ 1qpy´1q dt
“
ż 8
0
pt` 1qp´x´yq tpy´1qdt “
ż 8
0
tpy´1q
pt` 1qpx`yqdt.
(1.5)
Moreover, note that for all x P p0,8q it holds that
Γpx` 1q “
ż 8
0
tppx`1q´1q e´t dt “ ´
ż 8
0
tx“
´e´t‰
dt
“ ´
ˆ
“
txe´t‰t“8
t“0´ x
ż 8
0
tpx´1q e´t dt
˙
“ x
ż 8
0
tpx´1q e´t dt “ x ¨ Γpxq.
(1.6)
The proof of Lemma 1.1.2 is thus completed.
1.1.2 Monotonicity properties of the gamma and the betafunction
Lemma 1.1.3 (Montonicity property of the gamma function). It holds that
limxÑ8
Γ1pxq “ 8 (1.7)
and there exists a real number C P p0,8q such that for all x, y P rC,8q with x ď yit holds that Γpxq ď Γpyq.
1.1. PROPERTIES OF THE BETA AND THE GAMMA FUNCTION 17
Proof of Lemma 1.1.3. Observe that for all x P p0,8q it holds that
Γ1pxq “
ż 8
0
lnptq tpx´1q e´t dt
“
ż 1
0
lnptq tpx´1q e´t dt`
ż e
1
lnptq tpx´1q e´t dt`
ż 8
e
lnptq tpx´1q e´t dt
ě inftPp0,1q
“
lnptq tpx´1q e´t‰
`
ż 8
e
lnptq tpx´1q e´t dt
ě inftPp0,1q
“
lnptq tpx´1q‰
`
ż 8
e
tpx´1q e´t dt.
(1.8)
This proves that
limxÑ8
Γ1pxq ě inftPp0,1q
rlnptq ts ` limxÑ8
ż 8
e
tpx´1q e´t dt “ 8. (1.9)
The proof of Lemma 1.1.3 is thus completed.
Lemma 1.1.4 (Monotonicity of the beta function). For all x, y, x, y P p0,8q withx ď x and y ď y it holds that
Bpx, yq ď Bpx, yq. (1.10)
Proof of Lemma 1.1.4. Note that for all θ P p0, 1q, x, x P R with x ď x it holds that
θx ď θx. (1.11)
Combining this with Definition 1.1.1 completes the proof of Lemma 1.1.4.
1.1.3 Upper bounds for sums containing the beta and thegamma function
Lemma 1.1.5 (An upper bound for the beta function). Let x, y P p0,8q withpx´ 1q py ´ 1q ě 0. Then
Bpx, yq “
ż 1
0
p1´ tqpx´1q tpy´1q dt ď
ż 1
0
tpx`y´2q dt “
#
8 : x` y ď 11
px`y´1q: x` y ą 1
. (1.12)
18 CHAPTER 1. GRONWALL-TYPE INEQUALITIES
Proof of Lemma 1.1.5. First, observe that the equalities in (1.12) are clear. It thusremains to prove the inequality in (1.12). For this we assume w.l.o.g. that x` y ą 1,that x ‰ 1 and that y ‰ 1 (otherwise also the inequality in (1.12) is clear). Theassumption that px´ 1q py ´ 1q ě 0 hence shows that px´ 1q py ´ 1q ą 0 and thatpy´1qpx´1q
P p0,8q. Combining this with Holder’s inequality proves that
ż 1
0
p1´ tqpx´1q tpy´1q dt
ď
„ż 1
0
p1´ tqpx´1qr1`py´1qpx´1qs dt
´
r1`py´1qpx´1qs
´1¯
„ż 1
0
tpy´1qr1`px´1qpy´1q s dt
´
r1`px´1qpy´1q s
´1¯
“
„ż 1
0
p1´ tqpx`y´2q dt
´
r1`py´1qpx´1qs
´1¯
„ż 1
0
tpx`y´2q dt
´
r1`px´1qpy´1q s
´1¯
“
ż 1
0
tpx`y´2q dt.
(1.13)
The proof of Lemma 1.1.5 is thus completed.
Remark 1.1.6. Lemma 1.1.5, in particular, shows that for all x, y P p0,8q withpx´ 1q py ´ 1q ě 0 it holds that
Bpx, yq ď
ż 1
0
tpx`y´2q dt. (1.14)
However, it is not true that for all x, y P p0,8q it holds that
Bpx, yq ď
ż 1
0
tpx`y´2q dt. (1.15)
Indeed, observe that
limxŒ0
Bpx, 2´ xq “ limxŒ0
ż 1
0
p1´ tqpx´1q tp2´xq dt ě limxŒ0
ż 1
12
p1´ tqpx´1q tp2´xq dt
ě
„
1
2
p2´xq
limxŒ0
ż 1
12
p1´ tqpx´1q dt “ 8 ą 1 “
ż 1
0
tp2´2q dt.
(1.16)
Exercise 1.1.7. Prove that for all c P r0,8q, ε P p0,8q it holds that
8ÿ
n“1
cn
Γpnεqă 8. (1.17)
1.2. INTEGRAL OPERATORS RELATED TO THE BETA FUNCTION 19
Exercise 1.1.8. Prove that for all c P r0,8q, ε P p0,8q it holds that
8ÿ
n“1
cn„
nś
k“1
B`
kε, ε˘
ă 8. (1.18)
Exercise 1.1.9. Prove that for all c P r0,8q, ε, δ, ρ P p0,8q it holds that
8ÿ
n“1
cn„
n´1ś
k“0
B`
ε` kδ, ρ˘
ă 8. (1.19)
1.2 Integral operators related to the beta function
Lemma 1.2.1 (A scaling property of the beta function). For all β, γ P p0,8q,r, t P r0,8q with r ď t it holds that
ż t
r
pt´ sqpβ´1qps´ rqpγ´1q ds “ pt´ rqpβ`γ´1qBpβ, γq. (1.20)
Proof of Lemma 1.2.1. Note that for all β, γ P p0,8q, r, t P r0,8q with r ď t it holdsthat
ż t
r
pt´ sqpβ´1qps´ rqpγ´1q ds “
ż pt´rq
0
pt´ r ´ sqpβ´1q spγ´1q ds
“ pt´ rqpβ`γ´1q
ż 1
0
p1´ sqpβ´1q spγ´1q ds “ pt´ rqpβ`γ´1qBpβ, γq.
(1.21)
The proof of Lemma 1.2.1 is thus completed.
The next estimate, inequality (1.22) in Lemma 1.2.2, is an immediate consequenceof Lemma 1.2.1.
Lemma 1.2.2. Let α, γ, τ P R, T P rτ,8q, u PMpBprτ, T sq,Bpr0,8sqq, β P p0,8qsatisfy α ` γ ą 1. Then it holds for all t P rτ, T s that
ż t
τ
pt´ sqpβ´1qps´ τqpγ´1q upsq ds
ď pt´ τqpα`β`γ´2qB`
β, α ` γ ´ 1˘
«
supsPpτ,tq
upsq
ps´ τqpα´1q
ff
.
(1.22)
20 CHAPTER 1. GRONWALL-TYPE INEQUALITIES
We need a further estimate for the integral operator appearing on the left handside of (1.22). This is the subject of the next lemma.
Lemma 1.2.3 (Iterations of an integral operator). Let α, γ, τ P R, T P rτ,8q, b Pr0,8q, β P p0,8q, B : MpBprτ, T sq,Bpr0,8sqq Ñ MpBprτ, T sq,Bpr0,8sqq, assumethat mintα, βu`γ ą 1, and assume that for all u PMpBprτ, T sq,Bpr0,8sqq, t P rτ, T sit holds that
`
Bpuq˘
ptq “ b
ż t
τ
pt´ sqpβ´1qps´ τqpγ´1q upsq ds. (1.23)
Then it holds for all n P N, t P rτ, T s, u PMpBprτ, T sq,Bpr0,8sqq that`
Bnpuq
˘
ptq
ď bn pt´ τqpα´1`npβ`γ´1qq
«
n´1ź
k“0
B`
β, α ` γ ´ 1` kpβ ` γ ´ 1q˘
ff«
supsPpτ,tq
upsq
ps´τqpα´1q
ff
(1.24)
and that
`
Bnpuq
˘
ptq ď bn pt´ τqpn´1qrγ´1s`
«
n´1ź
k“1
B`
β, kpβ ´ r1´ γs`q˘
ff
¨t
∫τpt´ sqpβ`pn´1qpβ´r1´γs`q´1q
ps´ τqpγ´1q upsq ds.
(1.25)
Proof of Lemma 1.2.3. Estimate (1.24) is an immediate consequence of Lemma 1.2.2.It thus remains to prove estimate (1.25). For this we assume in the following w.l.o.g.that τ “ 0. Then note that Lemma 1.2.1 implies that for all u PMpBpr0, T sq,Bpr0,8sq,t P r0, T s, b P r0,8q, β, γ P p0,8q with β ` γ ą 1 it holds that
b
ż t
0
pt´ sqpβ´1q spγ´1q
„
b
ż s
0
ps´ rqpβ´1q rpγ´1q uprq dr
ds
“ b b
ż t
0
ż s
0
pt´ sqpβ´1q spγ´1qps´ rqpβ´1q rpγ´1q uprq dr ds
“ b b
ż t
0
rpγ´1q uprq
„ż t
r
pt´ sqpβ´1q spγ´1qps´ rqpβ´1q ds
dr
ď b b trγ´1s`ż t
0
rpγ´1q uprq
„ż t
r
pt´ sqpβ´1qps´ rqpβ´1`mintγ´1,0uq ds
dr
ď B`
β, β ´ r1´ γs`˘
b b trγ´1s`ż t
0
pt´ rqpβ`β´r1´γs`´1q rpγ´1q uprq dr.
(1.26)
1.3. GENERALIZED EXPONENTIAL-TYPE FUNCTIONS 21
Iterating (1.26) shows that for all u PMpBpr0, T sq,Bpr0,8sq, t P r0, T s, n P t2, 3, . . . uit holds that
`
Bnpuq
˘
ptq ď bn tpn´1qrγ´1s`t
∫0pt´ sqpβ`pn´1qpβ´r1´γs`q´1q spγ´1q upsq ds
¨
«
n´1ź
k“1
B`
β, kpβ ´ r1´ γs`q˘
ff
.
(1.27)
The proof of Lemma 1.2.3 is thus completed.
1.3 Generalized exponential-type functions
Definition 1.3.1 (Generalized exponential-type functions). We denote by Er : r0,8q Ñr0,8q, r P p0,8q, Er : r0,8q Ñ r0,8q, r P p0,8q, and Er : r0,8q Ñ r0,8q,r P p0,8q, the functions with the property that for all r P p0,8q, x P r0,8q itholds that
Errxs “8ÿ
n“0
xnr
Γpnr ` 1q, Errxs “ Er
”
pxΓprqq1r
ı
“
8ÿ
n“0
pxΓprqqn
Γpnr ` 1q(1.28)
and Errxs “b
Errx2s “
«
8ÿ
n“0
px2Γprqqn
Γpnr ` 1q
ff12
. (1.29)
1.4 Generalized Gronwall-type inequalities
Lemma 1.4.1 (Main idea in the proof of the generalized Gronwall inequality). Letτ P R, T P rτ,8q, b P MpBprτ, T s2q,Bpr0,8sqq, a, e P MpBprτ, T sq,Bpr0,8sqq,B : MpBprτ, T sq,Bpr0,8sqq ÑMpBprτ, T sq,Bpr0,8sqq satisfy that for all t P rτ, T s,u PMpBprτ, T sq,Bpr0,8sqq it holds that
`
Bpuq˘
ptq “t
∫τbpt, squpsq ds, (1.30)
and assume that e ď a`Bpeq. Then it holds for all n P N that
e ďn´1ÿ
k“0
Bkpaq `Bn
peq. (1.31)
22 CHAPTER 1. GRONWALL-TYPE INEQUALITIES
Proof of Lemma 1.4.1. Estimate (1.31) follows immediately from an iterated applica-tion of the assumption e ď a`Bpeq and from the fact that B is monotone in the sensethat for all u, u P MpBprτ, T sq,Bpr0,8sqq with u ď u it holds that Bpuq ď Bpuq.The proof of Lemma 1.4.1 is thus completed.
Next we present the generalized Gronwall inequalities. They are modified versionsof the estimates in Section 7.1 in Henry [10].
Theorem 1.4.2. Let τ P R, b P r0,8q, T P rτ,8q, a, e PMpBprτ, T sq,Bpr0,8sqq,β, γ P p0,8q, B : MpBprτ, T sq,Bpr0,8sqq ÑMpBprτ, T sq,Bpr0,8sqq satisfy β ` γ ą
1 andşT
τps´ τqpγ´1q epsq ds ă 8, assume that for all u P MpBprτ, T sq,Bpr0,8sqq,
t P rτ, T s it holds that
pBuqptq “ bt
∫τpt´ sqpβ´1q
ps´ τqpγ´1q upsq ds, (1.32)
and assume that for all t P rτ, T s it holds that
eptq ď aptq ` bt
∫τpt´ sqpβ´1q
ps´ τqpγ´1q epsq ds. (1.33)
Then it holds for all t P rτ, T s that
eptq ď8ÿ
n“0
`
Bnpaq
˘
ptq ď aptq `8ÿ
n“1
bn pt´ τqpn´1qrγ´1s`„
n´1ś
k“1
B`
β, kpβ ´ r1´ γs`q˘
¨t
∫τpt´ sqpβ`pn´1qpβ´r1´γs`q´1q
ps´ τqpγ´1q apsq ds (1.34)
and it holds for all t P pτ, T s, α P p0,8q with α ` γ ą 1 that
eptq ď8ÿ
n“0
`
Bnpaq
˘
ptq ď aptq (1.35)
`
«
supsPpτ,tq
apsq
ps´τqpα´1q
ff
8ÿ
n“1
bn pt´ τqpα´1`npβ`γ´1qq
„
n´1ś
k“0
B`
β, α ` γ ´ 1` kpβ ` γ ´ 1q˘
looooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooon
ă8
.
Proof of Theorem 1.4.2. W.l.o.g. we assume that τ “ 0. Lemma 1.4.1 implies thatfor all n P N0 it holds that
e ď a`Bpaq `B2paq ` . . .`Bn
paq `Bpn`1qpeq “
«
nÿ
k“0
Bkpaq
ff
`Bpn`1qpeq. (1.36)
1.4. GENERALIZED GRONWALL-TYPE INEQUALITIES 23
Next we note that inequality (1.25) in Lemma 1.2.3 together with the assumptionthat @ t P r0, T s :
şt
0spγ´1q epsq ds ă 8 and the fact that
@ c P r0,8q, r P p0,8q : limnÑ8
„
cn
Γppn´ 1qrq
“ 0 (1.37)
(see Exercise 1.1.7) implies that for all t P p0, T s it holds that
limnÑ8
`
Bnpeq
˘
ptq
ď limnÑ8
«
bn tβ´1`pn´1qpβ`γ´1qt
∫0spγ´1qepsq ds
«
n´1ź
k“1
B`
β, kpβ ´ r1´ γs`q˘
ffff
ď∫ t0 spγ´1q epsq ds
tγlimnÑ8
«
“
tpβ`γ´1q b‰n
«
n´1ź
k“1
B`
β, kpβ ´ r1´ γs`q˘
ffff
ď∫ t0 spγ´1q epsq ds
tγlimnÑ8
«
“
maxp1,Γpβqq tpβ`γ´1q b‰n
«
n´1ź
k“1
Γpkpβ´r1´γs`qqΓpβ`kpβ´r1´γs`qq
ffff
ď∫ t0 spγ´1q epsq ds
tγ
¨ limnÑ8
«
“
|1` Γpβq ` Γpβ ´ r1´ γs`q|2 tpβ`γ´1q b‰n
Γ`
β ` pn´ 1qpβ ´ r1´ γs`q˘
„
n´2ś
k“1
Γppk`1qpβ´r1´γs`qqΓpβ`kpβ´r1´γs`qq
ff
ď∫ t0 spγ´1q epsq ds
tγ
„
8ś
k“1
Γppk`1qpβ´r1´γs`qqΓpβ`kpβ´r1´γs`qq
¨ limnÑ8
«
“
|1` Γpβq ` Γpβ ´ r1´ γs`q|2 tpβ`γ´1q b‰n
Γ`
pn´ 1qpβ ´ r1´ γs`q˘
ff
“ 0.
(1.38)
This and (1.36) prove the first inequalities in (1.34) and (1.35). Estimate (1.25) inLemma 1.2.3 proves the second inequality in (1.34). Furthermore, estimate (1.24)in Lemma 1.2.3 proves the second inequality in (1.35). Finally, observe that Exer-cise 1.1.9 implies that for all t P pτ, T s it holds that
8ÿ
n“1
bn pt´ τqnpβ`γ´1q
„
n´1ś
k“0
B`
β, α ` γ ´ 1` kpβ ` γ ´ 1q˘
“
8ÿ
n“1
”
b pt´ τqpβ`γ´1qın
„
n´1ś
k“0
B`
β, α ` γ ´ 1` kpβ ` γ ´ 1q˘
ă 8.
(1.39)
The proof of Theorem 1.4.2 is thus completed.
24 CHAPTER 1. GRONWALL-TYPE INEQUALITIES
1.4.1 Gronwall-type inequalities with a singularity at the ini-tial time
The next result, Corollary 1.4.3, specialises estimate (1.34) in Theorem 1.4.2 to thecase where γ “ 1; see Lemma 7.1.1 in Henry [10].
Corollary 1.4.3. Let τ P R, b P r0,8q, T P rτ,8q, a, e PMpBprτ, T sq,Bpr0,8sqq,β P p0,8q satisfy
şT
τepsq ds ă 8 and assume that for all t P rτ, T s it holds that
eptq ď aptq ` bt
∫τpt´ sqpβ´1q epsq ds. (1.40)
Then it holds for all t P rτ, T s that
eptq ď aptq ` rΓpβq bs1β
ż t
τ
E1β“
pt´ sq rΓpβq bs1β‰
apsq ds. (1.41)
Proof of Corollary 1.4.3. Inequality (1.34) in Theorem 1.4.2 with γ “ 1 shows thatfor all t P rτ, T s it holds that
eptq ď aptq `
ż t
τ
«
8ÿ
n“1
rΓpβq bsn pt´ sqpnβ´1q
Γpnβq
ff
apsq ds
“ aptq ` rΓpβq bs1β
ż t
τ
«
8ÿ
n“1
“
pt´ sq rΓpβq bs1β‰pnβ´1q
Γpnβq
ff
apsq ds.
(1.42)
Next note that Lemma 1.1.2 shows that for all x P p0,8q it holds that
E1βpxq “8ÿ
n“1
nβxpnβ´1q
Γpnβ ` 1q“
8ÿ
n“1
xpnβ´1q
Γpnβq. (1.43)
Combining this with (1.42) completes the proof of Corollary 1.4.3.
The next result, Corollary 1.4.4, specialises estimate (1.35) in Theorem 1.4.2 tothe case where the function a in Theorem 1.4.2 satisfies aptq “ c tpα´1q for all t P pτ, T sand some c P r0,8q; see Exercise 3 in Henry [10]. Corollary 1.4.4 follows immediatelyfrom (1.35) in Theorem 1.4.2 and from Exercise 1.1.9.
1.4. GENERALIZED GRONWALL-TYPE INEQUALITIES 25
Corollary 1.4.4. Let τ P R, a, b P r0,8q, T P rτ,8q, α, β, γ P p0,8q, a, e P
MpBprτ, T sq,Bpr0,8sqq satisfy mintα, βu ` γ ą 1 andşT
τps´ τqpγ´1q epsq ds ă 8
and assume that for all t P pτ, T s it holds that
eptq ď a pt´ τqpα´1q` b
t
∫τpt´ sqpβ´1q
ps´ τqpγ´1q epsq ds. (1.44)
Then it holds for all t P pτ, T s that
eptq ď a pt´ τqpα´1q8ÿ
n“0
bn pt´ τqnpβ`γ´1q
„
n´1ś
k“0
B`
β, α ` γ ´ 1` kpβ ` γ ´ 1q˘
ă 8.
(1.45)
The next result, Corollary 1.4.5, specialises Corollary 1.4.4 to the case γ “ 1; cf.Exercise 4 in Henry [10]. Corollary 1.4.5 follows immediately from Corollary 1.4.4.
Corollary 1.4.5. Let τ P R, T P rτ,8q, e PMpBprτ, T sq,Bpr0,8sqq, a, b P r0,8q,α, β P p0,8q satisfy
şT
τepsq ds ă 8 and assume that for all t P pτ, T s it holds that
eptq ď a pt´ τqpα´1q` b
t
∫τpt´ sqpβ´1q epsq ds. (1.46)
Then it holds for all t P pτ, T s that
eptq ď a pt´ τqpα´1q8ÿ
n“0
bn pt´ τqnβ„
n´1ś
k“0
B`
β, α ` kβ˘
ă 8. (1.47)
1.4.2 Gronwall-type inequalities without a singularity at theinitial time
In the remainder of these lecture notes we will often use the following Gronwall-typeestimate; see Lemma 7.1.1 in Henry [10].
Corollary 1.4.6. Let τ P R, β P p0,8q, a, b P r0,8q, e PMpBprτ, T sq,Bpr0,8sqqsatisfy
şT
τepsq ds ă 8 and assume that for all t P rτ, T s it holds that
eptq ď a` bt
∫τpt´ sqpβ´1q epsq ds. (1.48)
Then it holds for all t P rτ, T s that
eptq ď a ¨ Eβ
”
pt´ τq pbΓpβqq1βı
“ a ¨ Eβ
“
pt´ τqβ b‰
. (1.49)
26 CHAPTER 1. GRONWALL-TYPE INEQUALITIES
Proof of Corollary 1.4.6. Corollary 1.4.3 implies that for all t P rτ, T s it holds that
eptq ď a` rΓpβq bs1β
ż t
τ
E1β“
pt´ sq rΓpβq bs1β‰
a ds. (1.50)
The fundamental theorem of calculus hence shows that for all t P rτ, T s it holds that
eptq ď a
„
1` limεŒ0
ż t
τ`ε
rΓpβq bs1β E1β
“
ps´ τq rΓpβq bs1β‰
ds
“ a
„
1` limεŒ0
“
Eβ
“
pt´ τq rΓpβq bs1β‰
´ Eβ
“
ε rΓpβq bs1≉
“ a“
1``
Eβ
“
pt´ τq rΓpβq bs1β‰
´ Eβ
“
0‰˘‰
“ aEβ
“
pt´ τq rΓpβq bs1β‰
.
(1.51)
The proof of Corollary 1.4.6 is thus completed.
Chapter 2
Regularity of nonlinear functions
Most of this chapter is based on Da Prato & Zabczyk [7] and Prevot & Rockner [24].
2.1 General functions
Definition 2.1.1 (Power set). Let A be a set. Then we denote by PpAq the powerset of A (the set of all subsets of A).
Definition 2.1.2 (Set of functions). Let A and B be sets. Then we denote byMpA,Bq the set of all functions from A to B.
Definition 2.1.3 (Domain of definition, range/codomain, image of a function). LetA and B be sets and let f : AÑ B be a function. Then
• we denote by Dpfq the set given by Dpfq “ A and we call Dpfq the domain ofdefinition of f ,
• we denote by rangepfq the set given by rangepfq “ B and we call rangepfq therange/codomain of f ,
• we denote by impfq the set given by impfq “ fpAq and we call impfq the imageof f , and
• we denote by graphpfq the set given by graphpfq “ tpx, yq P AˆB : y “ fpxquand we call graphpfq the graph of f .
27
28 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS
2.2 Measurable functions
We first recall the notion of a measurable mapping.
Definition 2.2.1 (Measurable functions). Let pΩ1,F1q and pΩ2,F2q be measurablespaces and let f : Ω1 Ñ Ω2 be a function with the property that for all F P F2 it holdsthat
f´1pF q P F1. (2.1)
Then f is called F1/F2-measurable.
Definition 2.2.2 (Measurable functions). Let pΩ1,F1q and pΩ2,F2q be measurablespaces. Then we denote by MpF1,F2q the set of all F1/F2-measurable functions.
Definition 2.2.3 (Borel sigma-algebra). Let pE, Eq be a topological space. Then wedenote by BpEq the set given by BpEq “ σEpEq and we call BpEq the Borel sigma-algebra on pE, Eq.
Definition 2.2.4 (Distance of sets). Let pE, dEq be a metric space. Then we denoteby distE : PpEq ˆ PpEq Ñ r0,8s the function with the property that for all A,B P
PpEq it holds that
distEpA,Bq “
#
infaPA infbPB dEpa, bq : A ‰ H and B ‰ H
8 : else. (2.2)
2.2.1 Nonlinear characterization of the Borel sigma-algebra
The next proposition shows that if pE, dEq is a metric space, then BpEq is the smallestsigma-algebra with the property that every continuous real-valued function is Borelmeasurable.
Proposition 2.2.5 (Nonlinear characterization of the Borel sigma-algebra). LetpE, dEq be a metric space. Then
BpEq “ σE`
pϕqϕPCpE,Rq˘
“ σEpϕ : ϕ P CpE,Rqq
“ σE`
ϕ´1pAq P PpEq : ϕ P CpE,Rq, A P BpRq
(˘
.(2.3)
2.2. MEASURABLE FUNCTIONS 29
Proof of Proposition 2.2.5. First of all, observe that for every ϕ P CpE,Rq it holdsthat ϕ is BpEq/BpRq-measurable. Hence, we obtain that
BpEq Ě σE`
ϕ´1pAq P PpEq : ϕ P CpE,Rq, A P BpRq
(˘
. (2.4)
It thus remains to prove that
BpEq Ď σE`
ϕ´1pAq P PpEq : ϕ P CpE,Rq, A P BpRq
(˘
. (2.5)
For this observe by that definition it holds that
BpEq “ σEptA P PpEq : A is an open set in pE, dEquq . (2.6)
It thus remains to prove that
tA P PpEq : A is an open set in pE, dEqu
Ď σE`
ϕ´1pAq P PpEq : ϕ P CpE,Rq, A P BpRq
(˘
.(2.7)
For this let B Ă E be an open set in pE, dEq and let ψ : E Ñ R be the function withthe property that for all x P E it holds that
ψpxq “ distEptxu, EzBq. (2.8)
Then it holds that ψ P CpE,Rq and this implies that
B “ ψ´1pp0,8qq Ď σE
`
ϕ´1pAq P PpEq : ϕ P CpE,Rq, A P BpRq
(˘
. (2.9)
The proof of Proposition 2.2.5 is thus completed.
Exercise 2.2.6. Let pΩ,Fq be a measurable space, let pE, dEq be a metric space, andlet f : Ω Ñ E be a function. Prove that f is F/BpEq-measurable if and only if itholds for all ϕ P CpE,Rq that ϕ ˝ f is F/BpRq-measurable.
2.2.2 Pointwise limits of measurable functions
Definition 2.2.7 (Sets of numbers). We denote by
N “ t1, 2, 3, . . . u (2.10)
the set of natural numbers, we denote by
N0 “ NY t0u “ t0, 1, 2, . . . u (2.11)
the union of t0u and the set of natural numbers, we denote by
Z “ t0, 1,´1, 2,´2, . . . u (2.12)
the set of integer numbers, we denote by Q the set of rational numbers, we denoteby R the set of real numbers and we denote by C the set of complex numbers.
30 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS
Note thatN Ă N0 Ă Z Ă Q Ă R Ă C. (2.13)
Lemma 2.2.8. Let pΩ,Fq be a measurable space, let Y : Ω Ñ R be a function, andlet Xn : Ω Ñ R, n P N, be a sequence of F/BpRq-measurable mappings such that forall ω P Ω it holds that Y pωq “ supnPNXnpωq. Then Y is F/BpRq-measurable.
Proof of Lemma 2.2.8. Note that for all c P R it holds that
tY ď cu “
"
supnPN
Xn ď c
*
“č
nPN
tXn ď culoooomoooon
PF
P F . (2.14)
The proof of Lemma 2.2.8 is thus completed.
Lemma 2.2.9. Let pΩ,Fq be a measurable space, let Y : Ω Ñ R be a function, andlet Xn : Ω Ñ R, n P N, be a sequence of F/BpRq-measurable mappings such that forall ω P Ω it holds that limnÑ8Xnpωq “ Y pωq. Then Y is F/BpRq-measurable.
Proof of Lemma 2.2.9. Note that Lemma 2.2.8 implies that for all c P R it holdsthat
tY ě cu “!
limnÑ8
Xn ě c)
“
"
lim supnÑ8
Xn ě c
*
“
#
limnÑ8
«
supmPtn,n`1,... u
Xm
ff
ě c
+
“č
nPN
#«
supmPtn,n`1,... u
Xm
ff
ě c
+
looooooooooooooomooooooooooooooon
PF
P F . (2.15)
The proof of Lemma 2.2.9 is thus completed.
The next corollary is an immediate consequence of Exercise 2.2.6 and Lemma 2.2.9;see, e.g., Proposition E.1 in [2] and Proposition A.1.3 in Prevot & Rockner [24].
Corollary 2.2.10. Let pΩ,Fq be a measurable space, let pE, dEq be a metric space,and let f : Ω Ñ E be a function. Then f is F/BpEq-measurable if and only if thereexists a sequence gn : Ω Ñ E, n P N, of F/BpEq-measurable functions such that forall ω P Ω it holds that
limnÑ8
dEpfpωq, gnpωqq “ 0. (2.16)
2.2. MEASURABLE FUNCTIONS 31
2.2.3 Lp-sets of measurable functions for p P r0,8q
Definition 2.2.11 (Lp-sets for p P r0,8q). Let K P tR,Cu, let pΩ,A, µq be ameasure space, let q P p0,8q, and let pV, ¨V q be a normed K-vector space. Then we
denote by L0pµ; ¨V q the set given by
L0pµ; ¨V q “MpA,BpV qq, (2.17)
we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the mapping with the property that
for all f P L0pµ; ¨V q it holds that
fLqpµ;¨V q“
„ż
Ω
fpωqqV µpdωq
1q
, (2.18)
and we denote by Lqpµ; ¨V q the set given by
Lqpµ; ¨V q “
f P L0pµ; ¨V q : fLqpµ;¨V q
ă 8(
. (2.19)
Definition 2.2.12 (Lp-sets for p P r0,8q). Let K P tR,Cu, p P r0,8q, q P p0,8q,let pΩ,A, µq be a measure space, and let pV, ¨V q be a normed K-vector space. Then
we denote by r¨sL0pµ;¨V q: L0pµ; ¨V q Ñ P
`
L0pµ; ¨V q˘
the function with the property
that for all f P L0pµ; ¨V q it holds that
rf sL0pµ;¨V q“
!
g P L0pµ; ¨V q : pDA P F : µpAq “ 0 and tf ‰ gu Ď Aq
)
, (2.20)
we denote by Lppµ; ¨V q the set given by
Lppµ; ¨V q “
rf sLppµ;¨V qĎ Lppµ; ¨V q : f P L
ppµ; ¨V q
(
, (2.21)
and we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function with the property
that for all f P L0pµ; ¨V q it holds that
›
›rf sLqpµ;¨V q
›
›
Lqpµ;¨V q“ fLqpµ;¨V q
. (2.22)
In the setting of Definition 2.2.12 we do in the following not distinguish betweenan element f P Lppµ; ¨V q of Lppµ; ¨V q and its equivalence class in Lppµ; ¨V q.
32 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS
2.3 Simple functions
The idea of the Lebesgue integral for real valued functions is to approximate thefunction by suitable simpler functions and then to define the Lebesgue integral ofthe “complicated” function as the limit of the integrals of the simpler functions.
Definition 2.3.1 (Simple functions). Let pΩ1,F1q and pΩ2,F2q be measurable spacesand let f : Ω1 Ñ Ω2 be an F1/F2-measurable function with the property that the setfpΩ1q is finite. Then f is called F1/F2-simple.
2.4 Strongly measurable functions
2.4.1 Separability
(Unfortunately) Measurable functions can, in general, not be approximated pointwise(see (2.16) in Corollary 2.2.10) by simple functions; see Theorem 2.4.7 below fordetails. To overcome this difficulty, we introduce the notion of a strongly measurablefunction; see Definition 2.4.3 below. For this the next definition is needed.
Definition 2.4.1 (Separability). A topological space pE, Eq is called separable if thereexist an at most countable set A Ď E such that A “ E.
A topological space that is not separable is in a certain sense extremely large.This, in turn, can cause several serious difficulties in the analysis of such spaces. Anexample of a non-separable topological space can be found below. Let a, b P R witha ă b. Then the R-Banach space pCpra, bs,Rq, ¨Cpra,bs,Rqq of continuous functions
from the interval ra, bs to R equipped with the supremum norm ¨Cpra,bs,Rq is anexample of a separable topological space as the set
"
f P Cpr0, 1s,Rq :
ˆ
Dn P N0 : Dλ0, . . . , λn P Q : @x P r0, 1s : fpxq “nř
k“0
λkxk
˙*
(2.23)is a countable dense subset of Cpr0, 1s,Rq.
Lemma 2.4.2. Let pE, dEq be a separable metric space and let A Ď E. Then themetric space pA, dE|AˆAq is separable.
Proof of Lemma 2.4.2. W.l.o.g. we assume that A ‰ H. Let penqnPN Ď E be asequence of elements in E such that the set ten P E : n P Nu is dense in E. In the
2.4. STRONGLY MEASURABLE FUNCTIONS 33
next step let pfnqnPN Ď A be a sequence of elements in A such that for all n P N itholds that
dEpfn, enq ď
#
0 : en P A
distEpA, tenuq `1
2n: en R A
. (2.24)
Then the set tfn P A : n P Nu is dense in A. Indeed, if v P A X ten P E : n P Nu,then
distEptvu, tf1, f2, . . . uq “ 0 (2.25)
and if v P Azten P E : n P Nu, then it holds for all n P N that
distEptvu, tf1, f2, . . . uq ď distEptvu, tfn, fn`1, . . . uq
“ infmPtn,n`1,... u
dEpv, fmq
ď infmPtn,n`1,... u
rdEpv, emq ` dEpem, fmqs
ď infmPtn,n`1,... u
„
dEpv, emq ` distEpA, temuq `1
2m
ď infmPtn,n`1,... u
„
2 dEpv, emq `1
2m
ď 2
„
infmPtn,n`1,... u
dEpv, emq
`1
2n
“ 2 distEptvu, ten, en`1, . . . uq `1
2n“
1
2n.
(2.26)
The proof of Lemma 2.4.2 is thus completed.
2.4.2 Strongly measurable functions
Definition 2.4.3 (Strongly measurable functions). Let pΩ,Fq be a measurable spaceand let pE, dEq be a metric space. A function f : Ω Ñ E is called strongly F/pE, dEq-measurable (or simply: strongly measurable) if f : Ω Ñ E is F/BpEq-measurable andif pfpΩq, dE|fpΩqˆfpΩqq is separable.
Let pΩ,Fq be a measurable space and let pE, dEq be a separable metric space.Lemma 2.4.2 then shows that every F/BpEq-measurable mapping f : Ω Ñ E is alsostrongly F/pE, dEq-measurable.
Exercise 2.4.4. Give an example of a measurable space pΩ,Fq, of a metric spacepE, dEq, and of an F/BpEq-measurable function f : Ω Ñ E which is not stronglyF/pE, dEq-measurable. Show that f is F/BpEq-measurable but not strongly F/pE, dEq-measurable.
34 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS
2.4.3 Pointwise approximations of strongly measurable func-tions
As mentioned above, measurable functions can, in general, not be approximatedpointwise by simple functions. However, strongly measurable functions can be ap-proximated pointwise by simple functions. This is the subject of the Theorem 2.4.7below (cf., e.g., Lemma 1.1 in Da Prato & Zabczyk [7] and Lemma A.1.4 in Prevot& Rockner [24]). In the proof of Theorem 2.4.7 the following two lemmas are used.
Lemma 2.4.5 (Projections in metric spaces). Let pE, dEq be a metric space, letn P N, let e1, . . . , en P E and let Ppe1,...,enq : E Ñ E be the function with the propertythat for all x P E it holds that
Ppe1,...,enqpxq “ emintkPt1,2,...,nu : dEpek,xq“distEptxu,te1,...,enuqu. (2.27)
Then Ppe1,...,enq is BpEq/PpEq-measurable and for all x P E it holds that
dEpx, Ppe1,...,enqpxqq “ distEptxu, te1, . . . , enuq. (2.28)
Proof of Lemma 2.4.5. Identity (2.28) is an immediate consequence of (2.27). LetD “ pD1, . . . , Dnq : E Ñ Rn be the function with the property that for all x P E itholds that
Dpxq “ pD1pxq, . . . , Dnpxqq “ pdEpx, e1q, . . . , dEpx, enqq . (2.29)
Observe that D is continuous and hence that D is BpEq/BpRnq-measurable. Thisimplies that for all k P t1, 2, . . . , nu it holds that
P´1pe1,...,enq
pekq “
x P E : Ppe1,...,enqpxq “ ek(
“
!
x P E : k “ min
l P t1, 2, . . . , nu : dEpel, xq “ distptxu, te1, . . . , enuq(
)
“
"
x P E : k “ min
"
l P t1, 2, . . . , nu : Dlpxq “ minuPt1,...,nu
Dupxq
**
“
"
x P E :
ˆ
Dkpxq ď minuPt1,...,nu
Dupxq and Dkpxq ă minuPt1,...,k´1u
Dupxq
˙*
“
"
x P E :
ˆ
@ l P t1, . . . , k ´ 1u : Dkpxq ă Dlpxq and@ l P t1, . . . , nu : Dkpxq ď Dlpxq
˙*
“
»
—
–
k´1č
l“1
tx P E : Dkpxq ă Dlpxquloooooooooooooomoooooooooooooon
PBpEq
fi
ffi
fl
č
»
—
–
nč
l“1
tx P E : Dkpxq ď Dlpxquloooooooooooooomoooooooooooooon
PBpEq
fi
ffi
fl
P BpEq.
(2.30)
2.4. STRONGLY MEASURABLE FUNCTIONS 35
This, in turn, implies that for all A P PpEq it holds that
P´1pe1,...,enq
pAq “ P´1pe1,...,enq
`
AX te1, . . . , enu˘
“ YfPAXte1,...,enu P´1pe1,...,enq
pfqlooooomooooon
PBpEq
P BpEq. (2.31)
The proof of Lemma 2.4.5 is thus completed.
Lemma 2.4.6. Let pΩ,Fq be a measurable space, let pE, dEq be a metric space andlet f : Ω Ñ E be a function, and let gn : Ω Ñ E, n P N, be a sequence of stronglyF/pE, dEq-measurable functions such that for all ω P Ω it holds that
limnÑ8
dEpfpωq, gnpωqq “ 0. (2.32)
Then f is strongly F/pE, dEq-measurable.
Proof of Lemma 2.4.6. Corollary 2.2.10 ensures that f is F/BpEq-measurable. Itthus remains to prove that fpΩq is separable. This follows from Lemma 2.4.2 andfrom the fact that YnPNgnpΩq is separable. The proof of Lemma 2.4.6 is thus com-pleted.
We now present the promised pointwise approximation result for strongly mea-surable functions.
Theorem 2.4.7 (Approximations of strongly measurable functions). Let pΩ,Fq bea measurable space, let pE, dEq be a metric space, and let f : Ω Ñ E be a function.Then the following four statements are equivalent:
(i) It holds that f is F/pE, dEq-strongly measurable.
(ii) There exists a sequence gn : Ω Ñ E, n P N, of strongly F/pE, dEq-measurablefunctions with the property that for all ω P Ω it holds that
limnÑ8
dEpfpωq, gnpωqq “ 0. (2.33)
(iii) There exists a sequence gn : Ω Ñ E, n P N, of simple functions with the prop-erty that for all ω P Ω it holds that
limnÑ8
dEpfpωq, gnpωqq “ 0. (2.34)
(iv) There exists a sequence gn : Ω Ñ E, n P N, of simple functions with the prop-erty that for all ω P Ω it holds that dEpfpωq, gnpωqq P r0,8q, n P N, decreasesmonotonically to zero.
36 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS
Proof of Theorem 2.4.7. Clearly, (iv) implies (iii) and (iii) implies (ii). Lemma 2.4.6shows that (ii) implies (i). It thus remains to prove that (i) implies (iv). For this letf : Ω Ñ E be a strongly F/pE, dEq-measurable function. The fact that f is stronglyF/pE, dEq-measurable ensures that fpΩq is separable. Hence, there exists a sequencepenqnPN Ď fpΩq of elements in fpΩq with the property that
ten P fpΩq : n P Nu Ě fpΩq. (2.35)
In the next step let Ppe1,...,enq : E Ñ E, n P N, and gn : Ω Ñ E, n P N, be thefunctions with the property that for all x P E, n P N it holds that
Ppe1,...,enqpxq “ emintkPt1,2,...,nu : dEpek,xq“distEptxu,te1,...,enuqu (2.36)
andgn “ Ppe1,...,enq ˝ f. (2.37)
Lemma 2.4.5 and the fact that f is F/BpEq-measurable implies that for all n P N itholds that gn is F/BpEq-measurable. In addition, by definition it holds for all n P Nthat gnpΩq Ď te1, . . . , enu is a finite set. We hence get that for all n P N it holdthat gn is an F/BpEq-simple function. Moreover, note that (2.28) in Lemma 2.4.5ensures that for all ω P Ω, n P N it holds that
dEpfpωq, gnpωqq “ dE`
fpωq, Ppe1,...,enqpfpωqq˘
“ distEpfpωq, te1, . . . , enuq . (2.38)
This and the property that for all ω P Ω it holds that distEpfpωq, te1, e2, . . . uq “ 0imply that for all ω P Ω, n P N it holds that
dEpfpωq, gnpωqq ě dEpfpωq, gn`1pωqq and limnÑ8
dEpfpωq, gnpωqq “ 0. (2.39)
The proof of Theorem 2.4.7 is thus completed.
2.4.4 Sums of strongly measurable functions
The next result, Corollary 2.4.8, shows that the sum of two strongly measurable map-pings is again a strongly measurable mapping. Corollary 2.4.8 follows immediatelyfrom Theorem 2.4.7.
Corollary 2.4.8. Let pΩ,Fq be a measurable space, let pV, ¨V q be a normed vectorspace and let f, g : Ω Ñ V be strongly F/pV, ¨V q-measurable mappings. Then f `g : Ω Ñ V is strongly F/pV, ¨V q-measurable.
2.4. STRONGLY MEASURABLE FUNCTIONS 37
2.4.5 Lp-spaces of strongly measurable functions for p P r0,8q
Definition 2.4.9 (Lp-spaces for p P r0,8q). Let K P tR,Cu, q P p0,8q, let pΩ,A, µqbe a measure space, and let pV, ¨V q be a normed K-vector space. Then we denoteby L0pµ; ¨V q the set given by
L0pµ; ¨V q “ tf PMpΩ, V q : f is strongly A/pV, ¨V q-measurableu, (2.40)
we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the mapping with the property that
for all f P L0pµ; ¨V q it holds that
fLqpµ;¨V q“
„ż
Ω
fpωqqV µpdωq
1q
P r0,8s, (2.41)
and we denote by Lqpµ; ¨V q the set given by
Lqpµ; ¨V q “
f P L0pµ; ¨V q : fLqpµ;¨V q
ă 8(
. (2.42)
Corollary 2.4.8 proves, in the setting of Definition 2.4.9, that for all p P r0,8q itholds that Lppµ; ¨V q is a K-vector space.
Definition 2.4.10 (Lp-spaces for p P r0,8q). Let K P tR,Cu, p P r0,8q, q P p0,8q,let pΩ,A, µq be a measure space, and let pV, ¨V q be a normed K-vector space. Thenwe denote by r¨sL0pµ;¨V q
: L0pµ; ¨V q Ñ PpL0pµ; ¨V qq the function with the property
that for all f P L0pµ; ¨V q it holds that
rf sL0pµ;¨V q“
!
g P L0pµ; ¨V q : f ´ gL1pµ;¨V q
“ 0)
, (2.43)
we denote by Lppµ; ¨V q the set given by
Lppµ; ¨V q “
rf sL0pµ;¨V qĎ L0
pµ; ¨V q : f P Lppµ; ¨V q
(
, (2.44)
and we denote by ¨Lqpµ;¨V q: L0pµ; ¨V q Ñ r0,8s the function with the property
that for all f P L0pµ; ¨V q it holds that›
›rf sLqpµ;¨V q
›
›
Lqpµ;¨V q“ fLqpµ;¨V q
. (2.45)
In the setting of Definition 2.4.10 we do in the following not distinguish betweenan element f P Lppµ; ¨V q of Lppµ; ¨V q and its equivalence class in Lppµ; ¨V q.
Lemma 2.4.11. Let K P tR,Cu, p P r1,8q, let pΩ,F , µq be a measure space andlet pV, ¨V q be a K-Banach space. Then Lppµ; ¨V q is a K-Banach space too.
38 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS
Lemma 2.4.12. Let K P tR,Cu, p P r1,8q, let pΩ,F , µq be a finite measure space,and let pV, ¨V q be a normed K-vector space. Then it holds that
rf sLppµ;¨V q: f is an F/BpV q-simple function
(
(2.46)
is dense in Lppµ; ¨V q.
Proof of Lemma 2.4.12. Let f P Lppµ; ¨V q. Theorem 2.4.7 proves that there existsa sequence gn : Ω Ñ V , n P N, of F/BpV q-simple functions with the property thatfor all ω P Ω it holds that fpωq ´ gnpωqV , n P N, decreases monotonically to zero.Lebesgue’s theorem of dominated convergence hence proves that
limnÑ8
f ´ gnLppµ;¨V q“ lim
nÑ8
„ż
Ω
fpωq ´ gnpωqpV µpdωq
1p
“ 0. (2.47)
The proof of Lemma 2.4.12 is thus completed.
2.5 Continuous functions
2.5.1 Topological spaces
Definition 2.5.1 (Topology). Let E be a set and let E Ď PpEq be a set such thatH, E P E, such that for all A Ď E it holds that YAPAA P E, and such that for allA,B P E it holds that pAXBq P E. Then E is called a topology on E and pE, Eq iscalled a topological space.
Proposition 2.5.2 (Topology induced by a function). Let E be a set, let T Ď R bea set, and let d : E ˆ E Ñ T be a function. Then it holds that the set
"
A P PpEq :´
@ v P A :“
D ε P p0,8q : tu P E : dpv, uq ă εu Ď A‰
¯
*
(2.48)
is a topology on E.
Proof of Proposition 2.5.2. Throughout this proof let E Ď PpEq be the set given by
E “"
A P PpEq :´
@ v P A :“
D ε P p0,8q : tu P E : dpv, uq ă εu Ď A‰
¯
*
. (2.49)
First, we observe thatH, E P E . Next we note that for all A Ď E and all v P rYAPAAsthere exists a set A P A and a real number ε P p0,8q such that v P A and such that
2.5. CONTINUOUS FUNCTIONS 39
tu P E : dpv, uq ă εu Ď A. In particular, this implies that for all A Ď E and allv P rYAPAAs there exists a real number ε P p0,8q such that tu P E : dpv, uq ă εu ĎrYAPAAs. Hence, we obtain that for all A Ď E it holds that
rYAPAAs P E . (2.50)
In the next step we observe that for all A,B P E and all v P pAXBq there existsreal numbers εA, εB P p0,8q such that
tu P E : dpv, uq ă εAu Ď A and tu P E : dpv, uq ă εBu Ď B. (2.51)
Hence, we obtain that for all A,B P E and all v P pAXBq there exists real numbersεA, εB P p0,8q such that
u P E : dpv, uq ă mintεA, εBu(
Ď pAXBq . (2.52)
This proves that for all A,B P E it holds that pAXBq P E . The proof of Proposi-tion 2.5.2 is thus completed.
Proposition 2.5.2 above ensures that the designation in the next definition isreasonable.
Definition 2.5.3 (Topology induced by a function). Let E be a set, let T Ď R be aset, and let d : E ˆ E Ñ T be a function. Then we denote by τpdq Ď PpEq the setgiven by
τpdq “
"
A P PpEq :´
@ v P A :“
D ε P p0,8q : tu P E : dpv, uq ă εu Ď A‰
¯
*
(2.53)
and we call τpdq the topology induced by d.
Lemma 2.5.4 (Balls are open). Let E be a set, let T Ď R be a set, let d : EˆE Ñ Tbe a function with the property that @x, y, z P E : dpx, zq ď dpx, yq ` dpy, zq, and letε P p0,8q, v P E. Then it holds that
tu P E : dpv, uq ă εu P τpdq. (2.54)
Proof of Lemma 2.5.4. First, observe that for all u P E with dpv, uq ă ε and allw P E with dpu,wq ă 1
2rε´ dpv, uqs it holds that
dpv, wq ď dpv, uq ` dpu,wq ă dpv, uq ` 12rε´ dpv, uqs “ 1
2dpv, uq ` ε
2ă ε. (2.55)
40 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS
This implies that for all x P tu P E : dpv, uq ă εu there exists a real number δ P p0,8qsuch that
tu P E : dpx, uq ă δu Ď tu P E : dpv, uq ă εu . (2.56)
Hence, we obtain that tu P E : dpv, uq ă εu P τpdq. The proof of Lemma 2.5.4 is thuscompleted.
Proposition 2.5.5 (Convergence in the induced topology). Let E be a set, let d : EˆE Ñ r0,8q be a function with the property that @x P E : dpx, xq “ 0 and @x, y, z PE : dpx, zq ď dpx, yq ` dpy, zq, and let e : N0 Ñ E be a function. Then it holds thatlim supnÑ8 d
`
ep0q, epnq˘
“ 0 if and only if for all A P τpdq with ep0q P A there existsa natural number N P N such that for all n P tN,N ` 1, . . . u it holds that epnq P A.
Proof of Proposition 2.5.5. First of all, recall that lim supnÑ8 d`
ep0q, epnq˘
“ 0 ifand only if @ ε P p0,8q : DN P N : @n P tN,N ` 1, . . . u : d
`
ep0q, epnq˘
ă ε. Hence,we obtain that lim supnÑ8 d
`
ep0q, epnq˘
“ 0 if and only if for all ε P p0,8q thereexists a natural number N P N such that for all n P tN,N ` 1, . . . u it holds thatepnq P tu P E : dpep0q, uq ă εu. This and Lemma 2.5.4 complete the proof of Propo-sition 2.5.5.
2.5.2 Semi-metric spaces
Definition 2.5.6. Let E be a set and let d : E ˆ E Ñ r0,8q be a function with theproperty that for all x, y, z P E it holds that
(i) dpx, xq “ 0,
(ii) dpx, yq “ dpy, xq, and
(iii) dpx, zq ď dpx, yq ` dpy, zq.
Then we call d a semi-metric (on E) and we call pE, dq a semi-metric space.
Definition 2.5.7 (Globally bounded sets). Let pE, dq be a semi-metric space andlet A Ď E be a set with the property that @ e P E : supaPA dpa, eq ă 8. Then we saythat A is d-globally bounded (that A is globally bounded).
Lemma 2.5.8 (Globally bounded sets). Let pE, dq be a semi-metric space with E ‰H and let A Ď E be a set. Then A is d-globally bounded if and only if D e PE : supaPA dpa, eq ă 8.
2.5. CONTINUOUS FUNCTIONS 41
Definition 2.5.9 (Globally bounded functions). Let E be a set, let pF, dq be a semi-metric space, and let f : E Ñ F be a function with the property that fpEq is ad-globally bounded set. Then we say that f is d-globally bounded (that f is globallybounded).
2.5.3 Continuity properties of functions
Definition 2.5.10 (Continuous functions). Let pE1, E1q and pE2, E2q be topologicalspaces. Then we denote by CpE1, E2q the set of all continuous functions from E1 toE2.
Definition 2.5.11 (Uniformly continuous). Let pE, dEq and pF, dF q be semi-metricspaces and let f : E Ñ F be a function with the property that
@ ε P p0,8q : D δ P p0,8q : @x, y P E : ppdEpx, yq ă δq ñ pdF pfpxq, fpyqq ă εqq .(2.57)
Then we say that f is uniformly continuous (dE/dF -uniformly continuous).
Definition 2.5.12 (Holder continuous functions). Let pE, dEq and pF, dF q be semi-metric spaces and let r P p0,8q. Then we denote by |¨|C0,rpE,F q : MpE,F q Ñ r0,8s
the function with the property that for all f PMpE,F q it holds that
|f |C0,rpE,F q “ sup
ˆ
t0u Y
"
dF pfpxq, fpyqq
|dEpx, yq|r P p0,8s : px, y P E, dF pfpxq, fpyqq ą 0q
*˙
(2.58)and we denote by C0,rpE,F q the set given by
C0,rpE,F q “
!
f PMpE,F q : |f |C0,rpE,F q ă 8
)
. (2.59)
Definition 2.5.13 (Holder continuous functions). Let pE, dEq and pF, dF q be semi-metric spaces, let r P p0,8q, and let f P C0,rpE,F q. Then f is called an r-Holdercontinuous function (with respect to dE/dF )
Lemma 2.5.14. Let pV, ¨V q and pW, ¨W q be normed R-vector spaces, let U Ď V bean open set, let v P U , and let f : U Ñ W be a function which is Frechet differentiablein v. Then
lim suphŒ0
„
F pv ` hq ´ F pvqVhV
“ limεŒ0
suphPV zt0u,hV ďε
„
F pv ` hq ´ F pvqVhV
“ F 1pvqLpV,W q .
(2.60)
42 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS
2.5.4 Modulus of continuity
Definition 2.5.15 (Modulus of continuity). Let pE, dEq and pF, dF q be semi-metricspaces and let f : E Ñ F be a function. Then we denote by wdE ,dFf : r0,8s Ñ r0,8sthe function with the property that for all h P r0,8s it holds that
wdE ,dFf phq “ sup´
t0u Y!
dF`
fpxq, fpyq˘
P r0,8q :“
x, y P E with dEpx, yq ď h‰
)¯
(2.61)and we call wdE ,dFf the modulus of continuity of f .
Lemma 2.5.16 (Properties of the modulus of continuity). Let pE, dEq and pF, dF qbe semi-metric spaces and let f : E Ñ F be a function. Then
• wdE ,dFf is non-decreasing,
• f is dE/dF -uniformly continuous if and only if limhŒ0wdE ,dFf phq “ 0,
• f is dF -globally bounded if and only if wdE ,dFf p8q ă 8, and
• for all x, y P E it holds that dF`
fpxq, fpyq˘
ď wdE ,dFf
`
dEpx, yq˘
.
The proof of Lemma 2.5.16 is clear and therefore omitted.
Exercise 2.5.17. Give an example of a metric space pE, dq such that for all h Pr0,8s it holds that
wd,didEphq “
#
0 : h P r0, 1q
1 : h P r1,8s. (2.62)
Prove that your metric space has the desired properties.
2.5.5 Extensions of uniformly continuous functions
Lemma 2.5.18 (Uniformly continuous functions). Let pE, dEq and pF, dF q be semi-metric spaces, let f : E Ñ F be a uniformly continuous function, and let penqnPN Ď Ebe a Cauchy sequence. Then fpenq P F , n P N, is a Cauchy sequence too.
Proof of Lemma 2.5.18. The assumption that penqnPN is a Cauchy sequence and theassumption that f is uniformly continuous imply that
limNÑ8
supn,mPtN,N`1,... u
dF pfpenq, fpemqq ď limNÑ8
supn,mPtN,N`1,... u
wdE ,dFf pdEpen, emqq
ď limNÑ8
wdE ,dFf
`
supn,mPtN,N`1,... u dEpen, emq˘
“ 0.(2.63)
2.5. CONTINUOUS FUNCTIONS 43
This shows that fpenq P F , n P N, is a Cauchy sequence. The proof of Lemma 2.5.18is thus completed.
Proposition 2.5.19 (Extension of uniformly continuous functions). Let pE, dEq bea semi-metric space, let pF, dF q be a complete semi-metric space, let A Ď E, andlet f : A Ñ F be a uniformly continuous function. Then there exists a unique f PCpA, F q with the property that f |A “ f , it holds for all h P r0,8s that
wdE ,dFf phq ď wdE ,dFf
phq ď limεŒ0
wdE ,dFf ph` εq (2.64)
and it holds that f is uniformly continuous.
Proof of Proposition 2.5.19. The uniqueness of f is clear. It remains to prove theexistence of a function f with the desired properties. For this observe that for allx P A and all penqnPN Ď A, penqnPN Ď A with limnÑ8 en “ x “ limnÑ8 en it holdsthat
lim supnÑ8
d`
fpenq, fpenq˘
ď lim supnÑ8
wdE ,dFf
`
dpen, enq˘
“ 0. (2.65)
This, Lemma 2.5.18, and the assumption that pF, dF q is complete imply that thereexist a function f : AÑ F with the property that for all x P A and all penqnPN Ď Awith limnÑ8 en “ x it holds that
fpxq “ limnÑ8
fpenq. (2.66)
In the next step we note that the continuity of f implies that for all x P A it holdsthat fpxq “ fpxq. In the next step we show (2.64). The first inequality in (2.64)is clear. To prove the second inequality in (2.64) let h P r0,8s and let x0, y0 P Awith dEpx0, y0q ď h. Then there exist sequences pxnqnPN Ď A and pynqnPN Ď A withthe property that limnÑ8 xn “ x0 and limnÑ8 yn “ y0. This implies that for allε P p0,8q it holds that
dF`
fpx0q, fpy0q˘
“ dF
´
limnÑ8
fpxnq, limnÑ8
fpynq¯
“ limnÑ8
dF pfpxnq, fpynqq
“ lim infnÑ8
dF pfpxnq, fpynqq ď lim infnÑ8
wf`
dEpxn, ynq˘
ď wf`
dEpx0, y0q ` ε˘
.(2.67)
This proves the second inequality in (2.64). The second inequality in (2.64), inturn, shows that f is uniformly continuous. The proof of Proposition 2.5.19 is thuscompleted.
44 CHAPTER 2. REGULARITY OF NONLINEAR FUNCTIONS
Exercise 2.5.20. Specify a metric space pE, dEq, a complete metric space pF, dF q,a set A Ď E, and a uniformly continuous function f : A Ñ F such that the uniquefunction f P CpA, F q with f |A “ f satisfies wf ‰ wf (i.e., there exists an h P r0,8ssuch that wf phq ‰ wf phq).
Remark 2.5.21. Let pE, dEq be a semi-metric space, let pF, dF q be a complete semi-metric space, let A Ď E be a dense subset of E, and let f : A Ñ F be a uniformlycontinuous function. Proposition 2.5.19 then proves that there exists a unique f PCpE,F q with f |A “ f . In the following we often write, for simplicity of presentation,f instead of f .
Lemma 2.5.22. Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be semi-normed K-vector spaces, and let A : V Ñ W be a linear mapping. Then A is continuous if andonly if A is uniformly continuous.
The proof of Lemma 2.5.22 is clear and therefore omitted.
Chapter 3
Linear functions
3.1 Linear spaces
We first recall some notions regarding linear spaces (also known as vector spaces).
Definition 3.1.1 (Span in a vector space). Let K be a field, let V be a K-vectorspace, and let A Ď V . Then we denote by spanV pAq Ď V the set with the propertythat
spanV pAq “ t0u Y tλ1a1 ` . . .` λnan P V : n P N, a1, . . . , an P A, λ1, . . . , λn P Ku .(3.1)
Definition 3.1.2 (Generating system). Let V be a vector space and let A Ď V be aset with the property that spanV pAq “ V . Then A is called a generating system (inV ).
Definition 3.1.3 (Linearly independent). Let K be a field, let V be a K-vector space,and let A Ď V be a set with the property that for all n P N, λ1, . . . , λn P K and allpairwise different a1, . . . , an P A with λ1a1 ` . . .` λnan “ 0 it holds that
λ1 “ . . . “ λn “ 0. (3.2)
Then A is called linearly independent (in V ).
Definition 3.1.4 (Basis of a vector space). Let V be a vector space and let A Ď Vbe a linearly independent generating system in V . Then A is called a (Hamel) basisof V .
Theorem 3.1.5 (Every vector space has a basis1). Let V be a vector space. Thenthere exists a subset A Ď V such that A is a basis of V .
1if one believes in the axiom of choice at least
45
46 CHAPTER 3. LINEAR FUNCTIONS
3.2 An intermezzo on sums over possibly uncount-
able index sets
3.2.1 Fubini’s theorem in the case of non-sigma-finite mea-sure spaces
Definition 3.2.1 (Counting measure on a set). Let A be a set. Then we denote by#A : PpAq Ñ r0,8s the counting measure on A.
Example 3.2.2. It holds that
ż
r0,1s
ż
r0,1s
1txupyq#RpdyqBorelRpdxq “
ż 1
0
ÿ
yPr0,1s
1txupyq dx “
ż 1
0
#Rptxuq dx “ 1,
ż
r0,1s
ż
r0,1s
1txupyq BorelRpdxq#Rpdyq “ÿ
yPr0,1s
ż 1
0
1txupyq dx “ÿ
yPr0,1s
BorelRptyuq “ 0.
(3.3)
3.2.2 Nets
Definition 3.2.3 (Relation). Let A and B be arbitrary sets and let C Ď AˆB be asubset of AˆB. Then the triple pA,B,Cq is called a (binary) relation (on pA,Bq).
Definition 3.2.4. Let „ “ pA,B,Cq be a relation, let a P A, and let b P B. Thenwe write a „ b if and only if pa, bq P C.
Definition 3.2.5 (Function). Let pA,B,Cq be a relation with the property that forevery a P A there exists exactly one b P B such that pa, bq P C. Then pA,B,Cq iscalled a function.
Definition 3.2.6 (Preorder). Let X be a set and let ĺ be a relation on pX,Xq withthe property that
(i) @x P X : x ĺ x (Reflexivity) and
(ii) @x, y, z P X :`
px ĺ y and y ĺ zq ñ px ĺ zq˘
(Transitivity).
Then the pair pX,ĺq is called a preorder (on X).
3.2. AN INTERMEZZOON SUMS OVER POSSIBLY UNCOUNTABLE INDEX SETS47
Definition 3.2.7 (Directed set). Let pX,ĺq be a preorder with the property that
@x, y P X : D z P X : px ĺ z and y ĺ zq . (3.4)
Then pX,ĺq is called a directed set.
Definition 3.2.8 (Nets). Let pX,ĺq be a directed set, let pE, Eq be a topologicalspace, and let φ : X Ñ E be a function from X to E. Then φ is called a net (frompX,ĺq to pE, Eq).
Definition 3.2.9 (Convergence of a net). Let pX,ĺq be a directed set, let pE, Eq bea topological space, let e P E, and let φ : X Ñ E be a net with the property that forevery neighbourhood U Ď E of e there exists an f0 P X such that for all f P X withf0 ĺ f it holds that φpfq P U . Then φ is said to converge (with respect to pX,ĺq) toe, in symbols,
limfPpX,ĺq
φpfq “ e. (3.5)
3.2.2.1 Confinal sequences
See, e.g., the book of Heuser [Analysis I, Satz 44.7] for this section.
Definition 3.2.10 (Confinal sequence). Let pX,ĺq be a directed set and let xn P X,n P N, be sequence with the property that for all y P X there exists a natural numberN P N such that for all n P tN,N ` 1, . . . u it holds that
y ĺ xn. (3.6)
Then pxnqnPN is called confinal (in pX,ĺq).
Proposition 3.2.11 (Convergence of confinal sequences). Let pX,ĺq be a directedset, let pE, Eq be a topological space, let e P E, let φ : X Ñ E be a net which convergesto e, and let xn P X, n P N, be a confinal sequence. Then it holds that the sequenceφpxnq, n P N, converges to e.
Proof of Proposition 3.2.11. Let U P E be an open set with the property that e P U .The assumption that φ converges to e ensures that there exists an element y P Xwith the property that for all z P X with y ĺ z it holds that
φpzq P U. (3.7)
48 CHAPTER 3. LINEAR FUNCTIONS
In the next step we note that the assumption that xn, n P N, is confinal implies thatthere exists a natural number N P N such that for all n P tN,N ` 1, . . . u it holdsthat
y ĺ xn. (3.8)
Combining (3.7) and (3.8) proves that for all n P tN,N ` 1, . . . u it holds that
φpxnq P U. (3.9)
The proof of Proposition 3.2.11 is thus completed.
Proposition 3.2.12 (Convergence of nets). Let pX,ĺq be a directed set, let xn P X,n P N, be a confinal sequence, let pE, Eq be a topological space, let φ : X Ñ E be a net,and assume that for every confinal seqence yn P X, n P N, it holds that φpynq P E,n P N, is a convergent sequence. Then there exists an element e P E such that φconverges to e and such that for every confinal seqence yn P X, n P N, it holds thatφpynq P E, n P N, converges to e.
Proof of Proposition 3.2.12. Throughout this proof we denote by p¨qb p¨q : MpN, Xq2 ÑMpN, Xq the mapping with the property that for all y, z : NÑ X it holds that
py b zqn “
#
zn2 : n is even
ypn`1q2 : n is odd(3.10)
Next observe that for all confinal sequences y, z : N Ñ X it holds that y b z is aconfinal sequence. By assumption we hence obtain that for all confinal sequencesy, z : NÑ X it holds that
limnÑ8
φ pynq “ limnÑ8
φ`
py b zqn˘
“ limnÑ8
φ pznq . (3.11)
This proves that there exists an element e P E such that for every confinal sequencey : NÑ X it holds that
limnÑ8
φpynq “ e. (3.12)
We now complete the proof of Proposition 3.2.12 by a contradiction, that is, weassume that φ does not converge to e. Hence, there exists an open set U P E withthe property that e P U and with the property that for all y P X there exists anelement z P X such that
y ĺ z and φpzq R U. (3.13)
3.2. AN INTERMEZZOON SUMS OVER POSSIBLY UNCOUNTABLE INDEX SETS49
This proves, in particular, that there exists a function z : N Ñ X such that for alln P N it holds that
xn ĺ zn and φpznq R U. (3.14)
The assumption that xn P X, n P N, is confinal and the fact that @n P N : xn ĺ znproves that z : N Ñ X is confinal too. This and (3.12) contradict to (3.14). Theproof of Proposition 3.2.12 is thus completed.
Example 3.2.13 (cf. Exercise 6 a) in Heuser). Let a, b P R with a ă b, let X be theset given by
X “ tA P Ppra, bsq : #RpAq ă 8u , (3.15)
and let ĺ be the relation on X with the property that for all A,B P X it holds thatA ĺ B if and only if A Ď B. Then there exists no sequence x : N Ñ X which isconfinal in pX,ĺq. Indeed, observe that for every sequence x : N Ñ X and everyt P
`
ra, bszpYnPNxnq˘
there exists no N P N such that ttu ĺ xn.
3.2.3 Sums over possibly uncountable index sets
Definition 3.2.14. Let A be a set and let f : A Ñ r0,8s be a function. Then wedenote by
ř
aPA fpaq the extended real number in r0,8s with the property that
ÿ
aPA
fpaq “
ż
A
fpaq#Apdaq. (3.16)
Another way to define the sum in (3.16) above is to employ the concept of a net.This is illustrated in the following example.
Example 3.2.15 (Sums through nets). Let A be a set, let f : AÑ r0,8s be a func-tion, and let φ : tx P PpAq : #Apxq ă 8u Ñ r0,8s be the function with the propertythat for all finite subsets x Ď A of A it holds that
φpxq “ÿ
aPx
fpaq. (3.17)
Then it holds that the pair
ptx P PpAq : #Apxq ă 8u ,Ďq (3.18)
is a directed set and it holds that φ is a net which converges toř
aPA fpaq.
50 CHAPTER 3. LINEAR FUNCTIONS
3.2.4 Fubini for sums
Lemma 3.2.16. Let A be a set and let f : AÑ r0,8s be a function withř
aPA fpaq ă8. Then it holds that the set f´1pp0,8sq is at most countable.
Proof of Lemma 3.2.16. Monotonicity proves that for all ε P p0,8q it holds that
#A
`
f´1prε,8qq
˘
¨ ε ďÿ
aPA
fpaq ă 8. (3.19)
This shows that for all n P N it holds that the set f´1`
r1n,8q˘
is finite. This impliesthat the set
f´1`
p0,8q˘
“ YnPNf´1`
r1n,8q˘
(3.20)
is at most countable. The proof of Lemma 3.2.16 is thus completed.
Lemma 3.2.17 (Fubini for sums). Let A and B be sets and let f : A ˆ B Ñ r0,8sbe a function. Then
ÿ
aPA
ÿ
bPB
fpa, bq “ÿ
bPB
ÿ
aPA
fpa, bq “ÿ
pa,bqPAˆB
fpa, bq. (3.21)
Proof of Lemma 3.2.17. W.l.o.g. we assume that pA ˆ Bq ‰ H is a non-empty set.We prove that
ÿ
aPA
ÿ
bPB
fpa, bq “ÿ
pa,bqPAˆB
fpa, bq. (3.22)
Clearly, (3.22) implies (3.21). To prove (3.22), we distinguish between several cases.In the first case we assume that
ÿ
pa,bqPAˆB
fpa, bq ă 8. (3.23)
This assumption together with Lemma 3.2.16 implies that there exists a sequencepan, bnq P Aˆ B, n P N, such that for all px, yq P pAˆ Bqztpan, bnq : n P Nu it holdsthat fpx, yq “ 0. The theorem of Fubini hence proves that
ÿ
pa,bqPAˆB
fpa, bq “ÿ
pa,bqPtan : nPNuˆtbn : nPNu
fpa, bq
“
ż
fpa, bq #tan : nPNuˆtbn : nPNupda, dbq
“
ż ż
fpa, bq #tbn : nPNupdbq #tan : nPNupdaq
“ÿ
aPtan : nPNu
ÿ
bPtbn : nPNu
fpa, bq “ÿ
aPA
ÿ
bPB
fpa, bq.
(3.24)
3.3. HILBERT SPACES 51
This finishes the proof of (3.22) in the caseÿ
pa,bqPAˆB
fpa, bq ă 8. (3.25)
In the second case we assume thatÿ
aPA
ÿ
bPB
fpa, bq ă 8. (3.26)
Lemma 3.2.16 implies then that there exists an at most countable set A Ď A suchthat for all a P AzA it holds that
ř
bPB fpa, bq “ 0. Moreover, again Lemma 3.2.16implies that there exist at most countable sets Ba Ď B, a P A, such that for alla P A, b P BzBa it holds that fpa, bq “ 0. The theorem of Fubini hence shows that
ÿ
aPA
ÿ
bPB
fpa, bq “ÿ
aPA
ÿ
bPB
fpa, bq “ÿ
aPA
ÿ
bPBa
fpa, bq “ÿ
aPA
ÿ
bPpYaPABaq
fpa, bq
“ÿ
pa,bqPAˆpYaPABaq
fpa, bq “ÿ
pa,bqPAˆB
fpa, bq.(3.27)
The proof of Lemma 3.2.17 is thus completed.
3.3 Hilbert spaces
In the next step we recall some notions regarding Hilbert spaces.
Definition 3.3.1 (Orthogonal). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space and letA Ď H. Then A is called orthogonal (in pH, 〈¨, ¨〉H , ¨Hq/in H) if for all a, b P Awith a ‰ b it holds that
〈a, b〉H “ 0. (3.28)
Definition 3.3.2 (Orthonormal). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space and letA Ď H be orthogonal. Then A is called orthonormal (in pH, 〈¨, ¨〉H , ¨Hq/in H) iffor all a P A it holds that
aH “ 1. (3.29)
Definition 3.3.3 (Completeness). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space. A setA Ď H is called complete (in pH, 〈¨, ¨〉H , ¨Hq/in H) if spanHpAq is dense in H, i.e.,if
spanHpAq “ H. (3.30)
52 CHAPTER 3. LINEAR FUNCTIONS
Definition 3.3.4 (Orthonormal basis). Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space andlet A Ď H be a complete and orthonormal set in H. Then A is called an orthonormalbasis of pH, 〈¨, ¨〉H , ¨Hq/of H.
Question 3.3.5. Let pH, 〈¨, ¨〉H , ¨Hq be an Hilbert space and let A Ď H be anorthonormal basis of H. Is A then a basis of H?
Theorem 3.3.6 (Every Hilbert space has an orthonormal basis2). Let pH, 〈¨, ¨〉H , ¨Hqbe a Hilbert space. Then there exists an orthonormal basis A Ď H of H.
Proposition 3.3.7 (A characterization for separable Hilbert spaces). LetK P tR,Cuand let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space. Then H is separable if and only if thereexists an at most countable orthonormal basis A Ď H of H.
Proof of Proposition 3.3.7. W.l.o.g. we assume that H ‰ t0u. If A Ď H is an atmost countable orthonormal basis of H, then the set
tλ1a1 ` . . .` λnan : n P N, a1, . . . , an P A, λ1, . . . , λn P tx P K : Repxq, Impxq P Quu(3.31)
is an at most countable dense subset of H. This proves the “ð” direction in thestatement of Proposition 3.3.7. The “ñ” direction in in the statement of Proposi-tion 3.3.7 follows from an application of the Gram-Schmidt process.
Example 3.3.8. Let er : R Ñ R, r P R, be the functions with the property that forall r, x P R it holds that
erpxq “
#
1 : x “ r
0 : x ‰ r. (3.32)
Then note that ter P L2p#R; |¨|
Rq : r P Ru is an orthonormal basis of the Hilbert space
L2p#R; |¨|Rq. Proposition 3.3.7 hence proves that the Hilbert space L2p#R; |¨|
Rq is not
separable.
Lemma 3.3.9. Let n P N, x “ px1, . . . , xnq P Rn, p P r1,8q. Then
xLpp#t1,2,...,nu;|¨|Rq“ p|x1|
p` . . .` |xn|
pq1pď |x1| ` . . .` |xn| “ xL1p#t1,2,...,nu;|¨|Rq
.
(3.33)
2if one believes in the axiom of choice at least
3.3. HILBERT SPACES 53
Proof of Lemma 3.3.9. Throughout this proof let e1, . . . , en P Rn be the vectors given
by e1 “ p1, 0, . . . , 0q, e2 “ p0, 1, 0, . . . , 0q, . . . , en “ p0, . . . , 0, 1q. Next observe thatthe triangle inequality implies that
xLpp#t1,2,...,nu;|¨|Rq“
›
›
›
›
›
nÿ
k“1
xkek
›
›
›
›
›
Lpp#t1,2,...,nu;|¨|Rq
ď
nÿ
k“1
xkekLpp#t1,2,...,nu;|¨|Rq
“
nÿ
k“1
|xk| ekLpp#t1,2,...,nu;|¨|Rq“
nÿ
k“1
|xk| “ xL1p#t1,2,...,nu;|¨|Rq.
(3.34)
The proof of Lemma 3.3.9 is thus completed.
3.3.1 Best approximations and projections in Hilbert spaces
Theorem 3.3.10 (Best approximation in Hilbert spaces). Let pH, 〈¨, ¨〉H , ¨Hq be aHilbert space, let U Ď H be a closed subspace of H, and let v P H. Then there existsa unique u P U with the property that
u´ vH “ infwPU
w ´ vH . (3.35)
Theorem 3.3.11 (Projection in Hilbert spaces). Let pH, 〈¨, ¨〉H , ¨Hq be a Hilbertspace and let U Ď H be a closed subspace of H. Then there exists a unique P P LpHqwith the property that P pHq Ď U and with the property that for all v P H it holdsthat
P pvq ´ vH “ infwPU
w ´ vH . (3.36)
Definition 3.3.12 (Projection in Hilbert spaces). Let pH, 〈¨, ¨〉H , ¨Hq be a Hilbertspace and let U Ď H be a closed subspace of H. Then we denote by PU P LpHq theunique bounded linear operator from H to H with the property that PUpHq Ď U andwith the property that for all v P H it holds that
PUpvq ´ vH “ infwPU
w ´ vH (3.37)
and we call PU the projection of H on U .
54 CHAPTER 3. LINEAR FUNCTIONS
3.3.2 Examples of orthonormal bases
3.3.2.1 Trigonometric functions
Lemma 3.3.13 (Real and imaginary part of product of complex numbers). For allz1, z2 P C it holds that
Repz1 ¨ z2q “ Repz1q ¨Repz2q ´ Impz1q ¨ Impz2q , (3.38)
Impz1 ¨ z2q “ Repz1q ¨ Impz2q ` Impz1q ¨Repz2q . (3.39)
The proof of Lemma 3.3.13 is clear. The next lemma presents a well-knownidentity for the difference of two arguments of the cosine function.
Lemma 3.3.14. For all x, y P R it holds that
cosp2xq ´ cosp2yq “ 2 sinpy ´ xq sinpy ` xq. (3.40)
Proof of Lemma 3.3.14. Throughout this proof let ϕ, ϕ0,1, ϕ1,1 P C8pR2,Rq be the
functions with the property that for all x, y P R it holds that
ϕpx, yq “ cosp2xq ´ cosp2yq ´ 2 sinpy ´ xq sinpy ` xq,
ϕ0,1px, yq “`
B
Byϕ˘
px, yq, ϕ1,1px, yq “`
B2
BxByϕ˘
px, yq.(3.41)
Next observe that for all x, y P R it holds that
ϕpx, yq “ ϕpx, 0q `
ż y
0
ϕ0,1px, sq ds
“ ϕpx, 0q `
ż y
0
„
ϕ0,1p0, sq `
ż x
0
ϕ1,1pr, sq dr
ds
“ ϕpx, 0q `
ż y
0
ϕ0,1p0, sq ds`
ż y
0
ż x
0
ϕ1,1pr, sq dr ds.
(3.42)
Moreover, observe that for all x, y P R it holds that
ϕ0,1px, yq “ 2 sinp2yq ´ 2 cospy ´ xq sinpy ` xq ´ 2 sinpy ´ xq cospy ` xq, (3.43)
ϕ0,1p0, yq “ 2 sinp2yq ´ 4 sinpyq cospyq, (3.44)
3.3. HILBERT SPACES 55
ϕ1,1px, yq “ ´2 sinpy ´ xq sinpy ` xq ´ 2 cospy ´ xq cospy ` xq
` 2 cospy ´ xq cospy ` xq ` 2 sinpy ´ xq sinpy ` xq “ 0.(3.45)
Equation (3.44) and Lemma 3.3.13 imply that for all y P R it holds that
ϕ0,1p0, yq “ 2 psinpy ` yq ´ cospyq sinpyq ´ sinpyq cospyqq
“ 2`
Im`
eiy ¨ eiy˘
´Re`
eiy˘
¨ Im`
eiy˘
´ Im`
eiy˘
¨Re`
eiy˘˘
“ 0.(3.46)
This, (3.45) and (3.42) prove that for all x, y P R it holds that
ϕpx, yq “ ϕpx, 0q “ cosp2xq ´ 1´ 2 sinp´xq sinpxq “ cosp2xq ´ 1` 2 |sinpxq|2
“ cosp2xq ` |sinpxq|2 ´ |cospxq|2
“ Re`
eix ¨ eix˘
´“
Re`
eix˘
¨Re`
eix˘
´ Im`
eix˘
¨ Im`
eix˘‰
.
(3.47)
This and Lemma 3.3.13 imply that for all x, y P R it holds that ϕpx, yq “ 0. Theproof of Lemma 3.3.14 is thus completed.
3.3.2.2 Orthonormal basis in L2pBorelp0,1q; |¨|Rq
Proposition 3.3.15. The sets
p?
2 sinpnπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
, (3.48)
p?
2 cospnπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
Y
1(
, (3.49)
1(
Y
p?
2 sinp2nπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
Y
p?
2 cosp2nπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
,(3.50)
p?
2 sinppn´ 12qπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
(3.51)
are orthonormal in L2pBorelp0,1q; |¨|Rq.
56 CHAPTER 3. LINEAR FUNCTIONS
Proof of Proposition 3.3.15. First of all, note that for all n P N it holds thatż 1
0
cospnπxq dx “ 0. (3.52)
Next observe that integration by parts proves that for all n,m P N it holds thatż 1
0
sinpnπxq sinpmπxq dx
“
„
´ cospnπxq sinpmπxq
nπ
x“1
x“0
`mπ
nπ
ż 1
0
cospnπxq cospmπxq dx
“mπ
nπ
ż 1
0
cospnπxq cospmπxq dx.
(3.53)
In addition, observe that (3.38) together with (3.52) ensures that for all n,m P N itholds that
ż 1
0
sinpnπxq sinpmπxq dx “
ż 1
0
Im`
einπx˘
¨ Im`
eimπx˘
dx
“
ż 1
0
Re`
einπx˘
¨Re`
eimπx˘
´Re`
einπx ¨ eimπx˘
dx
“
ż 1
0
cospnπxq cospmπxq dx´
ż 1
0
Re`
eipn`mqπx˘
dx
“
ż 1
0
cospnπxq cospmπxq dx.
(3.54)
Putting (3.54) into (3.53) proves that for all n,m P N with n ‰ m it holds thatż 1
0
sinpnπxq sinpmπxq dx “
ż 1
0
cospnπxq cospmπxq dx “ 0. (3.55)
In addition, observe that (3.54) implies that for all n P N it holds that
1 “
ż 1
0
|sinpnπxq|2 ` |cospnπxq|2looooooooooooooomooooooooooooooon
“1
dx “ 2
ż 1
0
|sinpnπxq|2 dx. (3.56)
This, (3.54) and (3.55) imply that for all n,m P N it holds thatż 1
0
?2 sinpnπxq ¨
?2 sinpmπxq dx “
ż 1
0
?2 cospnπxq ¨
?2 cospmπxq dx
“
#
0 : n ‰ m
1 : n “ m.
(3.57)
3.3. HILBERT SPACES 57
This and (3.56) prove that the set (3.48) is orthonormal in L2pBorelp0,1q; |¨|Rq. Fur-thermore, note that for all n P N it holds that
ż 1
0
1 ¨?
2 cospnπxq dx “?
2
ż 1
0
cospnπxq dx “?
2
„
sinpnπxq
nπ
x“1
x“0
“?
2
„
sinpnπq ´ sinp0q
nπ
“ 0.
(3.58)
This, (3.57) and (3.56) show that the set (3.49) is an orthonormal set in L2pBorelp0,1q; |¨|Rq.In the next step we observe that (3.38) and (3.52) imply that for all n,m P N it holdsthat
ż 1
0
sinppn´ 12qπxq sinppm´ 12qπxq dx
“
ż 1
0
Im`
eipn´12qπx˘
¨ Im`
eipm´12qπx˘
dx
“
ż 1
0
Re`
eipn´12qπx˘
¨Re`
eipm´12qπx˘
dx´Re`
eipn´12qπx¨ eipm´12qπx
˘
dx
“
ż 1
0
cosppn´ 12qπxq cosppm´ 12qπxq dx´Re
ˆż 1
0
eipn`m´1qπx dx
˙
“
ż 1
0
cosppn´ 12qπxq cosppm´ 12qπxq dx.
(3.59)
Furthermore, integration by parts proves that for all n,m P N it holds thatż 1
0
sinppn´ 12qπxq sinppm´ 12qπxq dx
“
„
´ cosppn´ 12qπxq sinppm´ 12qπxq
pn´ 12qπ
x“1
x“0
`pm´ 12qπ
pn´ 12qπ
ż 1
0
cosppn´ 12qπxq cosppm´ 12qπxq dx
“pm´ 12qπ
pn´ 12qπ
ż 1
0
cosppn´ 12qπxq cosppm´ 12qπxq dx.
(3.60)
As above we combine (3.59) and (3.60) to obtain that the set (3.51) is orthonormalin L2pBorelp0,1q; |¨|Rq.
Definition 3.3.16 (Hausdorff space). Let pE, Eq be a topological space with the prop-erty that for all a, b P E with a ‰ b there exists A,B P E with a P A, b P B andAXB “ H. Then pE, Eq is called a Hausdorff space.
58 CHAPTER 3. LINEAR FUNCTIONS
Theorem 3.3.17 (Stone-Weierstrass). Let K P tR,Cu, let pE, Eq be a compactHausdorff space, and let A Ď CpE,Kq be a subalgebra of CpE,Kq such that
(i) @ v P A : v P A (A is a sub-*-algebra of CpE,Kq),
(ii) 1 P A (A is a sub-*-algebra of CpE,Kq with 1) and
(iii) @x, y P E, x ‰ y : D v P A : vpxq ‰ vpyq (A seperates points).
Then A is dense in CpE,Kq.
A proof of Theorem 3.3.17 in German language can, for example, be found inHeuser [11]. We illustrate Theorem 3.3.17 by the following result.
Proposition 3.3.18. Let S “ tpa, bq P R2 : a2 ` b2 “ 1u Ď R2, let arg : S Ñ r0, 2πqbe the function with the property that for all x P S it holds that
`
cospargpxqq, sinpargpxqq˘
“ x, (3.61)
and let A Ď CpS,Cq be the set given by
A “ď
NPN
"ˆ
Nř
n“´N
an ei n argpxq
˙
xPS
P CpS,Cq : a´N , a1´N , . . . , aN P C
*
. (3.62)
Then A is dense in CpS,Cq.
Proof of Proposition 3.3.18. We prove Proposition 3.3.18 through an application ofTheorem 3.3.17. For this we note that for all v P A it holds that v P A. Moreover,note that
`
ei 0 argpxq˘
xPS“`
e0˘
xPS“ 1 P A. (3.63)
Furthermore, observe that for all n,m P Z, x P S it holds that
ei n argpxq¨ eim argpxq
“ ei pn`mq argpxq (3.64)
This ensures that A is a subalgebra of CpS,Cq. In order to apply Theorem 3.3.17, itremains to verify that A separates points. For this observe that pei argpxqqxPS P CpS,Cqand that for all x, y P S with argpxq ą argpyq it holds that
ei argpxq
ei argpyq“ eipargpxq´argpyqq
‰ 1. (3.65)
This ensures that for all x, y P S with x ‰ y it holds that
ei argpxq‰ ei argpyq. (3.66)
We can thus apply Theorem 3.3.17 to obtain that A is dense in CpS,Cq. The proofof Proposition 3.3.18 is thus completed.
3.3. HILBERT SPACES 59
Corollary 3.3.19. It holds that the set!
`
1?2πeinx
˘
xPp0,2πqP L2
pBorelp0,2πq; |¨|Cq : n P Z)
(3.67)
is an orthonormal basis of L2pBorelp0,2πq; |¨|Cq.
Proof of Corollary 3.3.19. Observe that for all n,m P Z it holds that
ż 2π
0
1?2πeinx 1?
2πeimx dx “
1
2π
ż 2π
0
eipm´nqx dx
“
$
&
%
”
12πipm´nq
eipm´nqxıx“2π
x“0“ 0 : m ‰ n
1 : m “ n.
(3.68)
This proves that the set `
1?2πeinx
˘
xPp0,2πqP L2pBorelp0,2πq; |¨|Cq : n P Z
(
is orthonor-
mal in L2pBorelp0,2πq; |¨|Cq. Next we denote by
CP pr0, 2πs,Cq “ tf P Cpr0, 2πs,Cq : fp0q “ fp2πqu (3.69)
the set of all 2π-periodic continuous functions from r0, 2πs to C. Proposition 3.3.18implies that
span!
`
1?2πeinx
˘
xPp0,2πq: n P Z
)CP pr0,2πs,Cq
“ CP pr0, 2πs,Cq. (3.70)
This implies that
span!
`
1?2πeinx
˘
xPp0,2πq: n P Z
)L2pBorelp0,2πq;|¨|Cq
Ě CP pr0, 2πs,CqL2pBorelp0,2πq;|¨|Cq
“ L2pBorelp0,2πq; |¨|Cq.
(3.71)
The proof of Corollary 3.3.19 is thus completed.
Exercise 3.3.20. Prove that the sets!
pcospnxqqxPp0,πq P L2pBorelp0,πq; |¨|Rq : n P N
)
Y t1u (3.72)
and!
psinpnxqqxPp0,πq P L2pBorelp0,πq; |¨|Rq : n P N
)
(3.73)
are orthonormal bases of L2pBorelp0,πq; |¨|Rq.
60 CHAPTER 3. LINEAR FUNCTIONS
Proposition 3.3.21. The sets
p?
2 sinpnπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
, (3.74)
p?
2 cospnπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
Y
1(
, (3.75)
1(
Y
p?
2 sinp2nπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
Y
p?
2 cosp2nπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
,(3.76)
p?
2 sinppn´ 12qπxqqxPp0,1q P L2pBorelp0,1q; |¨|Rq : n P N
(
(3.77)
are orthonormal bases of L2pBorelp0,1q; |¨|Rq.
Proposition 3.3.22 (Haar functions). Let Hn,k : p0, 1q Ñ R, k P t1, 2, . . . , 2nu,n P N0, be the functions with the property that for all n P N0, k P t1, 2, . . . , 2nu,t P p0, 1q it holds that
Hn,kptq “”
1`k´12n
,k´12
2n
˘ptq ´ 1`k´12
2n,k
2n
˘ptqı
2n2. (3.78)
Then it holds that the set
1 P L2pBorelp0,1q; |¨|Rq
(
Y
Hn,k P L2pBorelp0,1q; |¨|Rq : k P t1, 2, . . . , 2
nu, n P N0
(
(3.79)is an orthonormal basis of pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq
, ¨L2pBorelp0,1q;|¨|Rqq.
Proof of Proposition 3.3.22. First of all, we observe that for all n1, n2 P N0, k1 P
t1, 2, . . . , 2n1u, k2 P t1, 2, . . . , 2n2u it holds that
ż 1
0
Hn1,k1ptq ¨Hn2,k2ptq dt “
#
1 : pn1, k1q “ pn2, k2q
0 : pn1, k1q ‰ pn2, k2q(3.80)
andż 1
0
Hn1,k1ptq dt “ 0. (3.81)
This proves that the set
1 P L2pBorelp0,1q; |¨|Rq
(
Y
Hn,k P L2pBorelp0,1q; |¨|Rq : k P t1, 2, . . . , 2
nu, n P N0
(
(3.82)
3.3. HILBERT SPACES 61
is orthonormal in pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq, ¨L2pBorelp0,1q;|¨|Rq
q. It thus
remains to prove that the set
1 P L2pBorelp0,1q; |¨|Rq
(
Y
Hn,k P L2pBorelp0,1q; |¨|Rq : k P t1, 2, . . . , 2
nu, n P N0
(
(3.83)is complete in pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq
, ¨L2pBorelp0,1q;|¨|Rqq. To prove this,
we assert that for all m P N it holds that for all l P t1, 2, . . . , 2mu it holds that
1p0,1q,R`
l´12m
, l2m
˘ P spanL2pBorelp0,1q;|¨|Rq
´
1(
Y
Hn,k : k P t1, 2, . . . , 2nu, n P t0, 1, . . . ,m´1u(
¯
.
(3.84)In the following we prove (3.84) by induction on m P N. The base case m “ 1 followsfrom the fact that for all l P t1, 2u it holds that
1p0,1q,R`
l´12, l2
˘ “ 12
”
1p0,1q,Rp0,1q ` p´1qpl´1q
¨H0,1
ı
. (3.85)
The induction step N Q m Ñ m ` 1 P N follows from the induction hypothesis andfrom the fact that for all m P N, l P t1, 2, . . . , 2m`1u it holds that
1p0,1q,R`
l´1
2m`1 ,l
2m`1
˘ “ 12
„
1p0,1q,R`
rl2s´12m
,rl2s
2m
˘ `p´1qpl´1q
2m2¨Hm,rl2s
. (3.86)
Induction hence proves (3.84). Clearly, (3.84) implies (3.83). The proof of Proposi-tion 3.3.22 is thus completed.
3.3.2.3 Transformations of orthonormal bases
Exercise 3.3.23 (Transformation of orthonormal bases). Let a, b, α, β P R witha ă b and α ă β, let en : pa, bq Ñ R, n P N, be functions such that the set ten PL2pBorelpa,bq; |¨|Rq : n P Nu is an orthonormal basis of L2pBorelpa,bq; |¨|Rq, and letfn : pα, βq Ñ R, n P N, be the functions with the property that for all n P N,x P pα, βq it holds that
fnpxq “
d
pb´ aq
pβ ´ αqen
ˆ
px´ αq pb´ aq
pβ ´ αq` a
˙
. (3.87)
Prove that the set tfn P L2pBorelpα,βq; |¨|Rq : n P Nu is an orthonormal basis ofL2pBorelpα,βq; |¨|Rq.
62 CHAPTER 3. LINEAR FUNCTIONS
3.4 Linear functions
In the section we particularly follow the presentations in Werner [29].
Definition 3.4.1 (Linear operators). Let K be a field, let V1 and V2 be K-vectorspaces, and let A : V1 Ñ V2 be a function/operator3 with the property that for allv, w P V1, λ P K it holds that
A pλv ` wq “ λAv ` Aw. (3.88)
Then A is called K-linear.
Definition 3.4.2. Let K be a field and let V1 and V2 be K-vector spaces. Then wedenote by LinpV1, V2q the set given by
LinpV1, V2q “ tA PMpV1, V2q : A is linearu (3.89)
(the set of all linear functions from V1 to V2/the set of all linear operators from V1
to V2).
Definition 3.4.3 (Linear operators on a vector space). Let K be a field, let V1 andV2 be K-vector spaces, let U Ď V1 be a vector subspace of V1, and let A P LinpU, V2q.Then A is called a linear operator from U on V1 to V2 (a linear operator on V1 toV2).
Definition 3.4.4 (Set of linear operators on a vector space). Let K be a field andlet V1 and V2 be K-vector spaces. Then we denote by LpV1, V2q the set given by
LpV1, V2q “ď
UĎV1 is a vectorsubspace of V1
LinpU, V2q (3.90)
(the set of linear operators on V1 to V2).
Definition 3.4.5 (Point spectrum of a linear operator). Let K P tR,Cu, let V be aK-vector space, and let A : DpAq Ď V Ñ V be a linear operator. Then we denote byσP pAq the set given by
σP pAq “!
λ P K :`
λ´ A : DpAq Ñ V is not injective˘
)
(3.91)
and we call σP pAq the point spectrum of A (the set of eigenvalues of A).
3A function from some possibly infinite dimensional normed vector space into some possiblyinfinite dimensional normed vector space is also often referred as an “operator”. We will also oftenuse this convention in the remainder of these lecture notes.
3.4. LINEAR FUNCTIONS 63
Definition 3.4.6 (Symmetric linear operators). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, and let A : DpAq Ď H Ñ H be a linear operator with theproperty that for all v, w P DpAq it holds that
〈Av,w〉H “ 〈v, Aw〉H . (3.92)
Then A is called symmetric.
Definition 3.4.7. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA : DpAq Ď H Ñ H be a linear operator.
• A is called nonnegative if
@ v P DpAq : 〈v,Av〉H P r0,8q. (3.93)
• A is called nonpositive if
@ v P DpAq : 〈v, Av〉H P p´8, 0s. (3.94)
• A is called strictly positive if
@ v P DpAqzt0u : 〈v,Av〉H P p0,8q. (3.95)
• A is called strictly negative if
@ v P DpAqzt0u : 〈v, Av〉H P p´8, 0q. (3.96)
3.4.1 Continuous linear functions on normed vector spaces
Definition 3.4.8 (Bounded linear functions/bounded linear operators). Let K P
tR,Cu and let pV1, ¨V1q and pV2, ¨V2q be normed K-vector spaces. Then we denoteby LpV1, V2q the set given by
LpV1, V2q “ LinpV1, V2q X CpV1, V2q (3.97)
(the set of all continuous linear operators from V1 to V2) and we denote by ¨LpV1,V2q :
LpV1, V2q Ñ r0,8q the function with the property that for all A P LpV1, V2q it holdsthat
ALpV1,V2q “ supvPV1zt0u
„
AvV2vV1
. (3.98)
64 CHAPTER 3. LINEAR FUNCTIONS
Definition 3.4.9. Let K P tR,Cu and let pV, ¨V q be a normed K-vector space.Then we denote by LpV1q the set given by
LpV1q “ LpV1, V1q (3.99)
(the set of all continuous linear operators from V1 to V1) and we denote by ¨LpV1q : LpV1q Ñ
r0,8q the function with the property that for all A P LpV1q it holds that
ALpV1q “ ALpV1,V1q . (3.100)
Lemma 3.4.10 (Completeness of the space of bounded linear operators). Let K P
tR,Cu, let pV1, ¨V1q be a normed K-vector space, and let pV2, ¨V2q be a K-Banachspace. Then pLpV1, V2q, ¨LpV1,V2qq is a K-Banach space.
Definition 3.4.11 (Topological dual space). Let K P tR,Cu and let pV, ¨V q be anormed K-vector space. Then we denote by pV 1, ¨V 1q the K-Banach space given by
pV 1, ¨V 1q “ pLpV,Kq, ¨LpV,Kqq. (3.101)
See Reed & Simon [25] and, e.g., Prevot & Rockner [24] for the next results.
Theorem 3.4.12 (Square root of a nonnegative and symmetric bounded linearoperator). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A P
LpHq be nonnegative and symmetric. Then there exists a unique nonnegative andsymmetric S P LpHq with the property that S2 “ A.
Definition 3.4.13 (Square root of a nonnegative and symmetric bounded linearoperator). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A P
LpHq be nonnegative and symmetric. Then we denote by A12 P LpHq the uniquenonnegative and symmetric bounded linear operator with the property that pA12q2 “
A.
Lemma 3.4.14. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA P LpHq. Then A˚A is nonnegative and symmetric.
Proof of Lemma 3.4.14. Note that for all v P H it holds that
〈v, A˚Av〉H “ 〈Av,Av〉H “ 〈A˚Av, v〉H “ 〈Av,Av〉H “ Av
2H ě 0. (3.102)
The proof of Lemma 3.4.14 is thus completed.
3.4. LINEAR FUNCTIONS 65
3.4.2 Compact operators on Banach spaces
Definition 3.4.15 (Compact operator). LetK P tR,Cu, let pV1, ¨V1q and pV2, ¨V2qbe K-Banach spaces, and let A P LpV1, V2q satisfy that for every bounded subsetB Ď V1 of V1 it holds that ApBq is a relatively compact subset of V2, i.e., it holdsthat ApBq is a compact subset of V2. Then A is called a compact operator.
Definition 3.4.16 (The space of compact operators). Let K P tR,Cu and letpV1, ¨V1q and pV2, ¨V2q be K-Banach spaces. Then we denote by KpV1, V2q the setgiven by
KpV1, V2q “ tA P LpV1, V2q : A is compactu (3.103)
(the set of all compact linear operators from V1 to V2).
Proposition 3.4.17. Let K P tR,Cu and let pV1, ¨V1q and pV2, ¨V2q be K-Banachspaces. Then
tA P LpV1, V2q : dimpimpAqq ă 8uLpV1,V2q
Ď KpV1, V2q. (3.104)
Proposition 3.4.18. Let K P tR,Cu and let pV1, ¨V1q and pV2, ¨V2q be K-Hilbertspaces. Then
tA P LpV1, V2q : dimpimpAqq ă 8uLpV1,V2q
“ KpV1, V2q. (3.105)
3.4.3 Nuclear operators on Banach spaces
3.4.3.1 Definition of Nuclear operators
Definition 3.4.19 (Rank-1 operators). Let K P tR,Cu, let pV, ¨V q and pW, ¨W qbe normed K-vector spaces, let v P V 1, and let w P W . Then we denote by
`
w bv˘
: V Ñ W the function with the property that for all u P V it holds that
pw b vqpuq “ vpuqw. (3.106)
66 CHAPTER 3. LINEAR FUNCTIONS
Definition 3.4.20 (Nuclear operator). Let K P tR,Cu, let pV, ¨V q and pW, ¨W qbe K-Banach spaces, and let A : V Ñ W be a K-linear operator with the property thatthere exist a sequence pvnqnPN Ď V 1 of elements in V 1 and a sequence pwnqnPN Ď Wof elements in W such that
8ÿ
n“1
vnV 1 wnW ă 8 (3.107)
and such that for all x P V it holds that
Ax “8ÿ
n“1
pwn b vnqpxq “8ÿ
n“1
vnpxqwn. (3.108)
Then A is called a nuclear operator.
Conditions (3.107) and (3.108) say something about how good a linear operatorcan be approximated through sums of linear operators with one-dimensional images.In addition, conditions (3.107) and (3.108) assert that a nuclear operator can bedecomposed into rank one operators in the sense of (3.107)–(3.108). Equation (3.108)is also referred to as a nuclear represenation of a nuclear operator.
Definition 3.4.21 (The normed vector space of nuclear operators). Let K P tR,Cuand let pV, ¨V q and pW, ¨W q be K-Banach spaces. Then we denote by L1pV,W qthe set given by
L1pV,W q “ tA P LinpV,W q : A is nuclearu (3.109)
(the set of all nuclear operators from V to W ) and we denote by ¨L1pV,W q: L1pV,W q Ñ
r0,8q the function with the property that for all A P L1pV,W q it holds that
AL1pV,W q“ inf
#
a P r0,8q :
„
D pvnqnPN Ď V 1 : D pwnqnPN Ď W :
´
a “ř8
n“1 vnV 1 wnW ă 8 and @x P V : Ax “ř8
n“1pwn b vnqpxq¯
+
. (3.110)
Definition 3.4.22. Let K P tR,Cu and let pV, ¨V q be a K-Banach space. Then wedenote by L1pV q the set given by L1pV q “ L1pV, V q and we denote by ¨L1pV q
: L1pV q Ñ
r0,8q the mapping with the property that for all A P L1pV q it holds that AL1pV q“
AL1pV,V q.
3.4. LINEAR FUNCTIONS 67
3.4.3.2 Relation of bounded linear operators and nuclear operators
Lemma 3.4.23. Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be K-Banach spaces,and let A P L1pV,W q. Then A P LpV,W q and it holds that
ALpV,W q ď AL1pV,W q. (3.111)
Proof of Lemma 3.4.23. Let ε P p0,8q be arbitrary. The assumption that A P
L1pV,W q ensures that there exist sequences pvnqnPN Ď V 1 and pwnqnPN Ď W suchthat
8ÿ
n“1
vnV 1 wnW ď ε` AL1pV,W qď ε`
8ÿ
n“1
vnV 1 wnW ă 8 (3.112)
and such that for all x P V it holds that
Ax “8ÿ
n“1
vnpxqwn. (3.113)
Then note that for all x P V it holds that
AxW “
›
›
›
›
›
8ÿ
n“1
vnpxqwn
›
›
›
›
›
W
ď
8ÿ
n“1
vnpxqwnW “
8ÿ
n“1
|vnpxq| wnW
ď
8ÿ
n“1
vnLpV,Rq xV wnW “
«
8ÿ
n“1
vnV 1 wnW
ff
xV
ď
”
AL1pV,W q` ε
ı
xV .
(3.114)
This proves that A P LpV,W q and that
ALpV,W q ď AL1pV,W q` ε. (3.115)
As ε P p0,8q was arbitrary, the proof of Lemma 3.4.23 is completed.
Lemma 3.4.23, in particular, proves that in the setting of Lemma 3.4.23 it holdsthat
pL1pV,W q, ¨L1pV,W qq Ď pLpV,W q, ¨LpV,W qq (3.116)
continuously.
68 CHAPTER 3. LINEAR FUNCTIONS
3.4.3.3 Structure of the space of nuclear operators
Lemma 3.4.24. Let K P tR,Cu and let pV, ¨V q and pW, ¨W q be K-Banach spaces.Then the pair pL1pV,W q, ¨L1pV,W q
q is a normed K-vector space.
Proof of Lemma 3.4.24. Lemma 3.4.23 implies that for all A P L1pV,W q it holdsthat AL1pV,W q
“ 0 if and only if A “ 0. Furthermore, it is clear that for all
A P L1pV,W q, λ P K it holds that λ ¨ A P L1pV,W q and that
λ ¨ AL1pV,W q“ |λ| ¨ AL1pV,W q
. (3.117)
It thus remains to prove that the sum of two nuclear operators from V to W is againa nuclear operator from V to W and that the triangle inequality holds. For this letA1, A2 P L1pV,W q, ε P p0,8q be arbitrary and let vin P V
1, n P N, i P t1, 2u, andwin P W , n P N, i P t1, 2u, satisfy that for all i P t1, 2u it holds that
8ÿ
n“1
›
›vin›
›
V 1
›
›win›
›
Wď ε` AiL1pV,W q
ď ε`8ÿ
n“1
›
›vin›
›
V 1
›
›win›
›
Wă 8 (3.118)
and that for all x P V , i P t1, 2u it holds that
Aix “8ÿ
n“1
vinpxqwin. (3.119)
This implies that8ÿ
n“1
2ÿ
i“1
›
›vin›
›
V 1
›
›win›
›
Wă 8 (3.120)
and that for all x P V it holds that
pA1 ` A2qx “8ÿ
n“1
2ÿ
i“1
vinpxqwin
“ v11pxqw
11 ` v
21pxqw
21 ` v
12pxqw
12 ` v
22pxqw
22 ` v
13pxqw
13 ` v
23pxqw
23 ` . . . .
(3.121)
Hence, we obtain that A1 ` A2 P L1pV,W q and that
A1 ` A2L1pV,W qď
8ÿ
n“1
2ÿ
i“1
›
›vin›
›
V 1
›
›win›
›
Wď A1L1pV,W q
` A2L1pV,W q` 2ε. (3.122)
As ε P p0,8q was arbitrary, the proof of Lemma 3.4.24 is completed.
3.4. LINEAR FUNCTIONS 69
3.4.3.4 Ideal property of the set of nuclear operators
Proposition 3.4.25. Let K P tR,Cu, let pV0, ¨V0q, pV1, ¨V1q, pW0, ¨W0q and
pW1, ¨W1q be K-Banach spaces, and let A P L1pV1,W1q, B1 P LpW1,W0q, B2 P
LpV0, V1q. Then it holds that B1AB2 P L1pV0,W0q and it holds that
B1AB2L1pV0,W0qď B1LpW1,W0q
AL1pV1,W1qB2LpV1,V0q . (3.123)
Proof of Proposition 3.4.25. Let ε P p0,8q be arbitrary. The assumption that A PL1pV1,W1q ensures that there exist pvnqnPN Ď pV1q
1 and pwnqnPN Ď W1 such that
8ÿ
n“1
vnV 11wnW1
ď ε` AL1pV1,W1qď ε`
8ÿ
n“1
vnV 11wnW1
ă 8 (3.124)
and such that for all x P V1 it holds that
Ax “8ÿ
n“1
vnpxqwn. (3.125)
Inequality (3.124) implies that
8ÿ
n“1
vnpB2p¨qqV 10B1pwnqW0
ď
8ÿ
n“1
vnV 11B2LpV0,V1q B1LpW1,W0q
wnW1
ď B2LpV0,V1q B1LpW1,W0q
”
ε` AL1pV1,W1q
ı
(3.126)
and equation (3.125) ensures that for all x P V0 it holds that
pB1AB2q pxq “8ÿ
n“1
vnpB2pxqqB1pwnq. (3.127)
This implies that B1AB2 is a nuclear operator from V0 to W0 and that
B1AB2L1pV0,W0qď B2LpV0,V1q B1LpW1,W0q
”
ε` AL1pV1,W1q
ı
. (3.128)
As ε P p0,8q was arbitrary, the proof of Proposition 3.4.25 is completed.
70 CHAPTER 3. LINEAR FUNCTIONS
3.4.3.5 Characterization of nuclear operators
The next simple lemma gives a characterization for nuclear operators and is animmediate consequence of the definition of a nuclear operator.
Lemma 3.4.26. Let K P tR,Cu, let pV, ¨V q and pW, ¨W q be K-Banach spaceswith V ‰ t0u and W ‰ t0u, and let A P LpV,W q. Then the following three state-ments are equivalent:
(i) It holds that A P L1pV,W q.
(ii) There exist panqnPN Ď R, pvnqnPN Ď V 1, pwnqnPN Ď W such that for all n P Nit holds that vnV 1 “ wnW “ 1, such that
ř8
n“1 |an| ă 8 and such that forall x P V it holds that
Ax “8ÿ
n“1
an vnpxqwn. (3.129)
(ii) There exist panqnPN Ď r0,8q, pvnqnPN Ď V 1, pwnqnPN Ď W such that for alln P N it holds that vnV 1 “ wnW “ 1, such that
ř8
n“1 an ă 8 and such thatfor all x P V it holds that
Ax “8ÿ
n“1
an vnpxqwn. (3.130)
3.4.4 Hilbert-Schmidt operators on Hilbert spaces
Definition 3.4.27. Let K P tR,Cu, let pH1, 〈¨, ¨〉H1, ¨H1
q and pH2, 〈¨, ¨〉H2, ¨H2
q
be K-Hilbert spaces, and let A P LpH1, H2q be a bounded linear operator with theproperty that there exist an orthonormal basis B Ď H1 of H1 such that
ÿ
bPB
Ab2H2ă 8. (3.131)
Then A is called Hilbert-Schmidt.
3.4.4.1 Independence of the orthonormal basis
Lemma 3.4.28. Let K P tR,Cu, let pH1, 〈¨, ¨〉H1, ¨H1
q and pH2, 〈¨, ¨〉H2, ¨H2
q beK-Hilbert spaces, let B1 Ď H1 be an orthonormal basis of H1, let B2 Ď H2 be anorthonormal basis of H2, and let A P LpH1, H2q. Then
ÿ
bPB1
Ab2H2“
ÿ
bPB2
A˚b2H1. (3.132)
3.4. LINEAR FUNCTIONS 71
Proof of Lemma 3.4.28. Observe thatÿ
bPB1
Ab2H2“
ÿ
bPB1
ÿ
bPB2
ˇ
ˇxb, AbyH2
ˇ
ˇ
2“
ÿ
bPB1
ÿ
bPB2
ˇ
ˇxA˚b, byH1
ˇ
ˇ
2
“ÿ
bPB2
ÿ
bPB1
ˇ
ˇxA˚b, byH1
ˇ
ˇ
2“
ÿ
bPB2
A˚b2H1.
(3.133)
The proof of Lemma 3.4.28 is thus completed.
Lemma 3.4.29 (Independence of the orthonormal bases). Let K P tR,Cu, letpH1, 〈¨, ¨〉H1
, ¨H1q and pH2, 〈¨, ¨〉H2
, ¨H2q be K-Hilbert spaces, let B1 Ď H1 and
B2 Ď H1 be orthonormal bases of H1, and let A P LpH1, H2q. Thenÿ
bPB1
Ab2H2“
ÿ
bPB2
Ab2H2. (3.134)
Proof of Lemma 3.4.29. Theorem 3.3.6 implies that there exists an orthonormal ba-sis B Ď H2 of H2. Lemma 3.4.28 then implies that
ÿ
bPB1
Ab2H2“
ÿ
bPB
A˚b2H1“
ÿ
bPB2
Ab2H2. (3.135)
The proof of Lemma 3.4.29 is thus completed.
The next result, Corollary 3.4.30, gives a characterization of Hilbert-Schmidtoperators and follows immediately from Lemma 3.4.29 above.
Corollary 3.4.30. Let K P tR,Cu, let pH1, 〈¨, ¨〉H1, ¨H1
q and pH2, 〈¨, ¨〉H2, ¨H2
q
be K-Hilbert spaces, and let A P LpH1, H2q. Then A is a Hilbert-Schmidt operator ifand only if for every orthonormal basis B Ď H1 of H1 it holds that
ÿ
bPB
Ab2H1ă 8. (3.136)
3.4.4.2 The Hilbert space of Hilbert-Schmidt operators
Definition 3.4.31 (The space of Hilbert-Schmidt operators). Let K P tR,Cu andlet pH1, 〈¨, ¨〉H1
, ¨H1q and pH2, 〈¨, ¨〉H2
, ¨H2q be K-Hilbert spaces. Then we denote
by L2pH1, H2q and HSpH1, H2q the set given by
L2pH1, H2q “ HSpH1, H2q “ tA P LpH1, H2q : A is Hilbert-Schmidtu (3.137)
(the set of all Hilbert-Schmidt operators from H1 to H2).
72 CHAPTER 3. LINEAR FUNCTIONS
In the next definition, Definition 3.4.32, we introduce a norm on a space of Hilbert-Schmidt operators. Lemma 3.4.29 above ensures that (3.138) in Definition 3.4.32 doesindeed make sense.
Definition 3.4.32 (A norm on the space of Hilbert-Schmidt operators). Let K P
tR,Cu and let pH1, 〈¨, ¨〉H1, ¨H1
q and pH2, 〈¨, ¨〉H2, ¨H2
q be K-Hilbert spaces. Thenwe denote by ¨L2pH1,H2q
“ ¨HSpH1,H2q: L2pH1, H2q Ñ r0,8q the function with the
property that for all A P L2pH1, H2q and all orthonormal bases B Ď H1 of H1 it holdsthat
AL2pH1,H2q“ AHSpH1,H2q
“
«
ÿ
bPB
Ab2H2
ff12
. (3.138)
3.4.4.3 Hilbert-Schmidt embeddings
Lemma 3.4.33 (Hilbert-Schmidt embeddings). Let rn P Rzt0u, n P N, be realnumbers with
ř8
n“11
|rn|2ă 8, let pH, 〈¨, ¨〉H , ¨Hq and pH, 〈¨, ¨〉H , ¨Hq be R-Hilbert
spaces with H Ď H continuously, and let en P H, n P N, be an orthonormal basis ofH which satisfies that rnen, n P N, is an orthonormal basis of H. Then
〈v, w〉H “8ÿ
n“1
〈en, v〉H 〈en, w〉H|rn|
2 . (3.139)
Proof. Note that for all n P N it holds that
〈en, v〉H “
⟨en,
8ÿ
m“1
〈em, v〉H em
⟩H
“
8ÿ
m“1
〈em, v〉H 〈en, em〉H
“ 〈en, v〉H 〈en, en〉H “ 〈en, v〉H en2H “
〈en, v〉H|rn|2
¨ rnen2H “
〈en, v〉H|rn|2
.
(3.140)
This implies that
〈v, w〉H “8ÿ
n“1
〈rnen, v〉H 〈rnen, w〉H “8ÿ
n“1
〈en, v〉H 〈en, w〉H|rn|
2 . (3.141)
The proof of Lemma 3.4.33 is thus completed.
3.5. DIAGONAL LINEAR OPERATORS ON HILBERT SPACES 73
3.5 Diagonal linear operators on Hilbert spaces
Definition 3.5.1 (Diagonal linear operators). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, and let A : DpAq Ď H Ñ H be a linear operator with theproperty that there exists an orthonormal basis B Ď H of H and a function λ : BÑ K
such that
DpAq “
#
v P H :ÿ
bPB
|λb|2|〈b, v〉H |
2ă 8
+
(3.142)
and such that for all v P DpAq it holds that
Av “ÿ
bPB
λb 〈b, v〉H b. (3.143)
Then we say that A is a diagonal linear operator.
Exercise 3.5.2 (Diagonal operators are densely defined). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonal
linear operator. Prove that A is densely defined, i.e., prove that DpAq “ H.
Exercise 3.5.3. Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let A : DpAq Ď H Ñ
H be a diagonal linear operator. Prove that A is symmetric.
Exercise 3.5.4 (The point spectrum of a diagonal operator). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator, i.e., assume that there exists an orthonormal basis B Ď H of H anda function λ : BÑ K such that
DpAq “
#
v P H :ÿ
bPB
|λb|2|〈b, v〉H |
2ă 8
+
(3.144)
and such that for all v P DpAq it holds that Av “ř
bPB λb 〈b, v〉H b. Prove thatσP pAq “ impλq.
74 CHAPTER 3. LINEAR FUNCTIONS
Proposition 3.5.5 (Regularity of diagonal linear operators). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a linear operator,let B Ď H be an orthonormal basis of H, and let λ : BÑ K be a function such that
DpAq “
#
v P H :ÿ
bPB
|λb|2|〈b, v〉H |
2ă 8
+
(3.145)
and such that for all v P DpAq it holds that Av “ř
bPB λb 〈b, v〉H b. Then
(i) A P LpHq if and only if λ P L8p#B; |¨|Kq (ô λ is a bounded function ô
impλq “ σP pAq is a bounded set) and in that case it holds that ALpHq “
λL8p#B;|¨|Kq“ supbPB |λb|,
(ii) A P L2pHq “ HSpHq if and only if λ P L2p#B; |¨|Kq (ô
ř
bPB |λb|2ă 8) and
in that case it holds that AL2pHq“ λL2p#B;|¨|
Kq““ř
bPB |λb|2‰12
, and
(iii) A P L1pHq if and only if λ P L1p#B; |¨|Kq (ô
ř
bPB |λb| ă 8) and in that caseit holds that AL1pHq
“ λL1p#B;|¨|Kq“ř
bPB |λb|.
Question 3.5.6. Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, let b : N Ñ H be aninjective mapping such that bpNq “ tb1, b2, . . . u Ď H is an orthonormal basis of H,let Ar : DpArq Ď H Ñ H, r P R, be linear operators with the property that for allr P R it holds that DpArq “
v P H :ř8
n“1 n2r |〈bn, v〉H |
2ă 8
(
and with the propertythat for all r P R, v P DpArq it holds that
Arv “8ÿ
n“1
nr 〈bn, v〉H bn. (3.146)
Specify three subsets S1 Ď R, S2 Ď R and S3 Ď R of the real numbers
• such that for all r P R it holds that Ar P HSpHq if and only if r P S1,
• such that for all r P R it holds that Ar P LpHq if and only if r P S2, and
• such that for all r P R it holds that Ar P L1pHq if and only if r P S3.
3.5.1 Laplace operators on bounded domains
In this section we give functional analytic descriptions of Laplace operators with suit-able boundary conditions and thereby present a few important examples of diagonallinear operators.
3.5. DIAGONAL LINEAR OPERATORS ON HILBERT SPACES 75
3.5.1.1 Laplace operators with Dirichlet boundary conditions
Definition 3.5.7 (Laplace operator with Dirichlet boundary conditions). Let en PL2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all n P N and Borelp0,1q-almost all x P
p0, 1q it holds that enpxq “?
2 sinpnπxq, and let A : DpAq Ď L2pBorelp0,1q; |¨|Rq ÑL2pBorelp0,1q; |¨|Rq be the linear operator with the property that
DpAq “
#
v P L2pBorelp0,1q; |¨|Rq :
8ÿ
n“1
n4ˇ
ˇ 〈en, v〉L2pBorelp0,1q;|¨|Rq
ˇ
ˇ
2
Ră 8
+
(3.147)
and with the property that for all v P DpAq it holds that
Av “8ÿ
n“1
´π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (3.148)
Then we refer to A as the Laplace operator with Dirichlet boundary conditions onL2pBorelp0,1q; |¨|Rq.
Proposition 3.5.8. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq. Then itholds that A is a diagonal linear operator, it holds that σP pAq “ t´π
2, ´4π2, ´9π2,´16π2, . . . u, it holds that
DpAq “ H2pp0, 1q;Rq
loooooomoooooon
Sobolev space
XH10 pp0, 1q;Rq
loooooomoooooon
Sobolev space
“
$
’
’
’
&
’
’
’
%
v P H2pp0, 1q;Rq : lim
xŒ0vpxq
looomooon
“vp0`q
“ limxÕ1
vpxqlooomooon
“vp1´q
“ 0
,
/
/
/
.
/
/
/
-
,
(3.149)
and it holds for all v P DpAq that Av “ v2.
76 CHAPTER 3. LINEAR FUNCTIONS
3.5.1.2 Laplace operators with Neumann boundary conditions
Definition 3.5.9 (Laplace operator with Neumann boundary conditions). Let en PL2pBorelp0,1q; |¨|Rq, n P N0, satisfy that for all n P N and Borelp0,1q-almost all
x P p0, 1q it holds that e0pxq “ 1 and enpxq “?
2 cospnπxq, and let A : DpAq ĎL2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the linear operator with the property that
DpAq “
#
v P L2pBorelp0,1q; |¨|Rq :
8ÿ
n“1
n4ˇ
ˇ 〈en, v〉L2pBorelp0,1q;|¨|Rq
ˇ
ˇ
2
Ră 8
+
(3.150)
and with the property that for all v P DpAq it holds that
Av “8ÿ
n“0
´π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (3.151)
Then we refer to A as the Laplace operator with Neumann boundary conditions onL2pBorelp0,1q; |¨|Rq.
Proposition 3.5.10. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Neumann boundary conditions on L2pBorelp0,1q; |¨|Rq. Then itholds that A is a diagonal linear operator, it holds that σP pAq “ t0, ´π2, ´4π2,´9π2, ´16π2, . . . u, it holds that
DpAq “
$
’
’
’
&
’
’
’
%
v P H2pp0, 1q;Rq : lim
xŒ0v1pxq
looomooon
“v1p0`q
“ limxÕ1
v1pxqlooomooon
“v1p1´q
“ 0
,
/
/
/
.
/
/
/
-
, (3.152)
and it holds for all v P DpAq that Av “ v2.
3.5. DIAGONAL LINEAR OPERATORS ON HILBERT SPACES 77
3.5.1.3 Laplace operators with periodic boundary conditions
Definition 3.5.11 (Laplace operator with periodic boundary conditions). Let en PL2pBorelp0,1q; |¨|Rq, n P Z, satisfy that for all n P N and Borelp0,1q-almost all x P p0, 1q
it holds that e0pxq “ 1, enpxq “?
2 sinp2nπxq and e´npxq “?
2 cosp2nπxq, and letA : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the linear operator with theproperty that
DpAq “
#
v P H :ÿ
nPZ
n4ˇ
ˇ 〈en, v〉L2pBorelp0,1q;|¨|Rq
ˇ
ˇ
2
Ră 8
+
(3.153)
and with the property that for all v P DpAq it holds that
Av “ÿ
nPZ
´4π2n2 〈en, v〉L2pBorelp0,1q;|¨|Rqen. (3.154)
Then we refer to A as the Laplace operator with periodic boundary conditions onL2pBorelp0,1q; |¨|Rq.
Proposition 3.5.12. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with periodic boundary conditions on L2pBorelp0,1q; |¨|Rq. Then itholds that A is a diagonal linear operator, it holds that σP pAq “ t0, ´4¨π2, ´4¨22 ¨π2,´4 ¨ 32 ¨ π2, ´4 ¨ 42 ¨ π2, . . . u, it holds that
DpAq “ H2P pp0, 1q;Rq
loooooomoooooon
Sobolev space
“
$
’
’
’
&
’
’
’
%
v P H2pp0, 1q;Rq : lim
xŒ0vpxq
looomooon
“vp0`q
“ limxÕ1
vpxqlooomooon
“vp1´q
, limxŒ0
v1pxqlooomooon
“v1p0`q
“ limxÕ1
v1pxqlooomooon
“v1p1´q
,
/
/
/
.
/
/
/
-
,
(3.155)
and it holds for all v P DpAq that Av “ v2.
78 CHAPTER 3. LINEAR FUNCTIONS
3.5.2 Spectral decomposition for a diagonal linear operator
Proposition 3.5.13 (The eigenspaces of diagonal linear operators). Let K P tR,Cube a field, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be adiagonal linear operator, i.e., assume that there exists an orthonormal basis B Ď Hand a function λ : BÑ K such that
DpAq “
#
v P H :ÿ
bPB
|λb 〈b, v〉H |2ă 8
+
(3.156)
and such that for all v P DpAq it holds that
Av “ÿ
bPB
λb 〈b, v〉H b. (3.157)
Then for all µ P σP pAq it holds that
Kernpµ´ Aq “ spantb P B : λb “ µu “ spanpλ´1pµqq (3.158)
Proof of Proposition 3.5.13. Let µ P σP pAq be arbitrary. We first prove that
Kernpµ´ Aq Ď spantb P B : λb “ µu. (3.159)
For this observe that for all
v P Kernpµ´ Aq “ tw P Dpµ´ Aq “ DpAq : pµ´ Aqw “ 0u (3.160)
it holds that
0 “ 02H “ pµ´ Aqv2H “
ÿ
bPB
|pµ´ λbq 〈b, v〉H |2“
ÿ
bPB
|µ´ λb|2|〈b, v〉H |
2
“ÿ
bPB,λb‰µ
|µ´ λb|2|〈b, v〉H |
2 .(3.161)
Hence, we obtain that for all v P Kernpµ ´ Aq and all b P B with λb ‰ µ it holdsthat 〈b, v〉H “ 0. This implies that for all v P Kernpµ´ Aq it holds that
v P”
spantb P B : λb ‰ µuıK
. (3.162)
This and the identity”
spantb P B : λb ‰ µuı
k
”
spantb P B : λb “ µuı
“ H (3.163)
3.5. DIAGONAL LINEAR OPERATORS ON HILBERT SPACES 79
prove that (3.159) is indeed fufilled. Next we prove that
Kernpµ´ Aq Ě spantb P B : λb “ µu. (3.164)
For this observe that for all
v P spantb P B : λb “ µu “”
spantb P B : λb ‰ µuıK
(3.165)
it holds thatÿ
bPB
|λb 〈b, v〉H |2“
ÿ
bPBλb“µ
|λb 〈b, v〉H |2“ |µ|2
ÿ
bPBλb“µ
|〈b, v〉H |2“ |µ|2 v2H ă 8. (3.166)
This shows that
spantb P B : λb “ µu “”
spantb P B : λb ‰ µuıK
Ď DpAq “ Dpµ´ Aq. (3.167)
Next note that for all
v P spantb P B : λb “ µu “”
spantb P B : λb ‰ µuıK
(3.168)
it holds that
pµ´ Aq v “ÿ
bPB
pµ´ λbq 〈b, v〉H b “ÿ
bPBλb“µ
pµ´ λbq 〈b, v〉H b “ 0. (3.169)
The proof of Proposition 3.5.13 is thus completed.
The next result, Theorem 3.5.14, establishes a spectral decomposition for diagonallinear operators. It follows immediately from Proposition 3.5.13 above.
Theorem 3.5.14 (Spectral decomposition for diagonal linear operators). Let K P
tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be adiagonal linear operator. Then it holds that
DpAq “
$
&
%
v P H :ÿ
λPσP pAq
|λ|2›
›PKernpλ´Aqpvq›
›
2
Hă 8
,
.
-
(3.170)
and it holds for all v P DpAq that
Av “ÿ
λPσP pAq
λ ¨ PKernpλ´Aqpvq. (3.171)
Exercise 3.5.15. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA : DpAq Ď H Ñ H be a diagonal linear operator. Prove that A is symmetric if andonly if σP pAq Ď R.
80 CHAPTER 3. LINEAR FUNCTIONS
3.5.3 Fractional powers of a diagonal linear operator
Definition 3.5.16 (Nonnegative fractional powers of a diagonal linear operator).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let r P r0,8q, and letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď r0,8q. Then wedenote by Ar : DpArq Ď H Ñ H the linear operator with the property that
DpArq “
$
&
%
v P H :ÿ
λPσP pAq
›
›λr ¨ PKernpλ´Aqpvq›
›
2
Hă 8
,
.
-
(3.172)
and with the property that for all v P DpArq it holds that
Arv “ÿ
λPσP pAq
λr ¨ PKernpλ´Aqpvq. (3.173)
Definition 3.5.17 (Negative fractional powers of a diagonal linear operator). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let r P p´8, 0q, and letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q. Then wedenote by Ar : DpArq Ď H Ñ H the linear operator with the property that
DpArq “
$
&
%
v P H :ÿ
λPσP pAq
›
›λr ¨ PKernpλ´Aqpvq›
›
2
Hă 8
,
.
-
(3.174)
and with the property that for all v P DpArq it holds that
Arv “ÿ
λPσP pAq
λr ¨ PKernpλ´Aqpvq. (3.175)
The next lemma collects a simple property of fractional powers of a diagonallinear operator. It follows immediately from Definition 3.5.16 and Definition 3.5.17.
Lemma 3.5.18 (Diagonality of fractional powers of a diagonal linear operators).Let K P tR,Cu, r P R, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq ĎH Ñ H be a diagonal linear operator with σP pAq Ď r0,8q, and assume that
`
r Pr0,8q or σP pAq Ď p0,8q
˘
. Then Ar is a diagonal linear operator.
3.5. DIAGONAL LINEAR OPERATORS ON HILBERT SPACES 81
3.5.4 Domain Hilbert space associated to a diagonal linearoperator
Lemma 3.5.19. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, letA : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q. Thenthe triple
`
DpAq, 〈Ap¨q, Ap¨q〉H , Ap¨qH˘
is a K-inner product space.
The proof of Lemma 3.5.19 is clear and therefore omitted. If the point spectrum ofthe diagonal linear operator A in Lemma 3.5.19 in addition satisfies infpσP pAqq ą 0,then the triple
`
DpAq, 〈Ap¨q, Ap¨q〉H , Ap¨qH˘
is even a K-Hilbert space. This is thesubject of the next lemma.
Lemma 3.5.20. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, andlet A : DpAq Ď H Ñ H be a diagonal linear operator with σP pAq Ď p0,8q andinfpσP pAqq ą 0. Then the triple
`
DpAq, 〈Ap¨q, Ap¨q〉H , Ap¨qH˘
is a K-Hilbert spaceand it holds for all v P DpAq that
vH ďAvH
infpσP pAqq. (3.176)
Proof of Lemma 3.5.20. First of all, note that for all v P DpAq it holds that
Av2H “ÿ
µPσP pAq
›
›µ ¨ PKernpµ´Aqpvq›
›
2
H“
ÿ
µPσP pAq
|µ|2 ¨›
›PKernpµ´Aqpvq›
›
2
H
ě
„
infµPσP pAq
|µ|2
»
–
ÿ
µPσP pAq
›
›PKernpµ´Aqpvq›
›
2
H
fi
fl “
„
infµPσP pAq
µ
2
v2H .
(3.177)
This proves (3.176). Due to Lemma 3.5.19, it remains to prove that the normedK-vector space
`
DpAq, Ap¨qH˘
is complete. For this let pvnqnPN Ď DpAq be aCauchy sequence in
`
DpAq, Ap¨qH˘
. Inequality (3.176) hence implies that pvnqnPNis a Cauchy sequence in pH, ¨Hq too. This and the fact that pH, ¨Hq is completeshows that there exists a vector v P H such that limnÑ8 vn ´ vH “ 0. Next note
82 CHAPTER 3. LINEAR FUNCTIONS
that for all n P N it holds that
lim infmÑ8
Apvn ´ vmq2H “ lim inf
mÑ8
»
–
ÿ
µPσP pAq
|µ|2›
›PKernpµ´Aqpvn ´ vmq›
›
2
H
fi
fl
ěÿ
µPσP pAq
|µ|2”
lim infmÑ8
›
›PKernpµ´Aqpvn ´ vmq›
›
2
H
ı
“ÿ
µPσP pAq
|µ|2„
›
›
›PKernpµ´Aq
´
vn ´ limmÑ8
vm
¯›
›
›
2
H
“ÿ
µPσP pAq
|µ|2”
›
›PKernpµ´Aqpvn ´ vq›
›
2
H
ı
.
(3.178)
This proves that v P DpAq and that limnÑ8 Apvn ´ vqH “ 0. The proof ofLemma 3.5.20 is thus completed.
Exercise 3.5.21. Give an example of an R-Hilbert space pH, 〈¨, ¨〉H , ¨Hq and adiagonal linear operator A : DpAq Ď H Ñ H such that σP pAq Ď p0,8q and such thatthe triple
`
DpAq, 〈Ap¨q, Ap¨q〉H , Ap¨qH˘
is not an R-Hilbert space.
3.5.5 Interpolation spaces associated to a diagonal linear op-erator
Theorem 3.5.22 (Completion). Let pE, dEq be a metric space. Then there exists a
complete metric space pF, dF q such that E Ď F , EF“ F and dF |EˆE “ dE.
Theorem 3.5.22 can be proved by considering the set of equivalence classes ofCauchy sequences in E. The detailed proof of Theorem 3.5.22 is well known andtherefore omitted.
Definition 3.5.23 (Completion). Let pE, dEq be a metric space and let pF, dF q be a
complete metric space such that E Ď F , EF“ F and dF |EˆE “ dE. Then the metric
space pF, dF q is called a completion of pE, dEq.
We now introduce the concept of a family of interpolation spaces associated to adiagonal linear operator.
3.5. DIAGONAL LINEAR OPERATORS ON HILBERT SPACES 83
Theorem 3.5.24 (Interpolation spaces associated to a diagonal linear operator). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ Hbe a symmetric diagonal linear operator with infpσP pAqq ą 0. Then there exists anup to isometric isomorphisms unique family pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, of K-Hilbertspaces with the property that
(i) @ r, s P R, r ě s : Hr Ď Hs “ HrHs
,
(ii) @ r P r0,8q : pDpArq, 〈Arp¨q, Arp¨q〉H , Arp¨qHq “ pHr, 〈¨, ¨〉Hr , ¨Hrq, and
(iii) @ r P p´8, 0s, v P H : vHr “ ArvH .
Proof of Theorem 3.5.24. Let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P r0,8q, be theK-Hilbert spaceswith the property that for all r P r0,8q it holds that
pHr, 〈¨, ¨〉Hr , ¨Hrq “ pDpArq, 〈Arp¨q, Arp¨q〉H , A
rp¨qHq. (3.179)
Note that Lemma 3.5.20 ensures that such K-Hilbert spaces do indeed exist. In thenext step let H8 be the set given by H8 “ XrPp0,8qHr. Then we observe that H8is a K-vector space. Furthermore, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P p´8, 0q, be K-Hilbertspaces with the property that for all r P p´8, 0q it holds that
Hr “
H Z
#
pH,ϕq P tHu ˆ LinpH8,Kq :
«
supvPH8zt0u
|ϕpvq|
vH´ră 8 “ sup
vPH8zt0u
|ϕpvq|
vH
ff+
,
(3.180)
with the property that for all r P p´8, 0q, λ P K, v, w P H it holds that
vHr “ supuPH8zt0u
|〈v, u〉H |uH´r
, v `Hr w “ v ` w, λ ¨Hr v “ λ ¨ v, (3.181)
with the property that for all r P p´8, 0q, λ P K, ϕ P LinpH8,Kq with supvPH8zt0u|ϕpvq|vH´r
ă 8 it holds that
pH,ϕqHr “ supuPH8zt0u
|ϕpuq|
uH´r, λ ¨Hr pH,ϕq “ pH, λ ¨ ϕq, (3.182)
84 CHAPTER 3. LINEAR FUNCTIONS
and with the property that for all r P p´8, 0q, v P H, ϕ, ψ P LinpH8,Kq with
supvPH8zt0u|ϕpvq|`|ψpvq|vH´r
ă 8 it holds that
pH,ϕq `Hr pH,ψq
“
#
w : rDw P H : @u P H8 : 〈w, u〉H “ ϕpuq ` ψpuqs
pH,ϕ` ψq : else
(3.183)
and
pH,ϕq `Hr v “ v `Hr pH,ϕq “´
H,“
H8 Q u ÞÑ 〈v, u〉H ` ϕpuq P K‰
¯
. (3.184)
The proof of Theorem 3.5.24 is thus completed.
Definition 3.5.25 (Interpolation spaces associated to a diagonal linear operator).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H bea symmetric diagonal linear operator with infpσP pAqq ą 0, and let pHr, 〈¨, ¨〉Hr , ¨Hrq,r P R, be K-Hilbert spaces with the property that
(i) @ r, s P R, r ě s : Hr Ď Hs “ HrHs
,
(ii) @ r P r0,8q : pDpArq, 〈Arp¨q, Arp¨q〉H , Arp¨qHq “ pHr, 〈¨, ¨〉Hr , ¨Hrq, and
(iii) @ r P p´8, 0s, v P H : vHr “ ArvH .
Then we call pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, a family of interpolation spaces associatedto A.
3.6 The Bochner integral
3.6.1 Existence and uniqueness of the Bochner integral
Theorem 3.6.1 (Bochner integral). Let pΩ,F , µq be a finite measure space, let K PtR,Cu, and let pV, ¨V q be a K-Banach space. Then
(i) there exists a unique continuous K-linear function I : L1pµ; ¨V q Ñ V with theproperty that for all F/BpV q-simple f : Ω Ñ V it holds that
Ipfq “ř
vPfpΩq
µpf´1ptvuqq ¨ v (3.185)
(ii) and it holds for all f P L1pµ; ¨V q that IpfqV ď fL1pµ;¨V q.
3.6. THE BOCHNER INTEGRAL 85
Proof of Theorem 3.6.1. Throughout this proof let S Ď L1pµ; ¨V q be the set of allF/BpV q-simple functions and let J : S Ñ V be the mapping with the property thatfor all f P S it holds that
Jpfq “ř
vPfpΩq
µpf´1ptvuqq ¨ v. (3.186)
Next observe that the triangle inequality proves that for all f P S it holds that
JpfqV ďř
vPfpΩq
µpf´1pvqq ¨ vV “ fL1pµ;¨V q. (3.187)
This, the fact that J is linear, and Lemma 2.5.22 imply that J is uniformly contin-uous. In addition, we note that item (iv) in Theorem 2.4.7 and Lebesgue’s theoremof dominated convergence ensure that SL1pµ;¨V q “ L1pµ; ¨V q. The assumption thatV is complete hence allows us to apply Proposition 2.5.19 to obtain that there existsa unique I P CpL1pµ; ¨V q, V q with the property that I|S “ J . This proves item (i).
In addition, observe that item (i), (3.187), and the fact that SL1pµ;¨V q “ L1pµ; ¨V qestablish (ii). The proof of Theorem 3.6.1 is thus completed.
3.6.2 Definition of the Bochner integral
Definition 3.6.2. Let pΩ,F , µq be a finite measure space, let K P tR,Cu, and letpV, ¨V q be a K-Banach space. Then we denote by
ş
Ωp¨q dµ : L1pµ; ¨V q Ñ V the
continuous K-linear function with the property that for all F/BpV q-simple f : Ω Ñ Vit holds that
ż
Ω
f dµ “ř
vPfpΩq
µpf´1ptvuqq ¨ v. (3.188)
Corollary 3.6.3 (Triangle inequality for the Bochner integral). Let pΩ,F , µq be afinite measure space, let K P tR,Cu, let pV, ¨V q be a K-Banach space, and letf P L1pµ; ¨V q. Then
›
›
›
›
ż
Ω
f dµ
›
›
›
›
V
ď
ż
Ω
fV dµ. (3.189)
Corollary 3.6.3 is an immediate consequence of Theorem 3.6.1.
86 CHAPTER 3. LINEAR FUNCTIONS
Chapter 4
Semigroups of bounded linearoperators
In this chapter we follow with some minor changes the presentations in Pazy [23].
4.1 Definition of a semigroup of bounded linear
operators
Definition 4.1.1 (Semigroups of bounded linear operators). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space, and let S : r0,8q Ñ LpV q be a mapping withthe property that for all t1, t2 P r0,8q it holds that
S0 “ IdV and St1St2 “ St1`t2looooooomooooooon
semigroup property
. (4.1)
Then we call S a semigroup (of bounded linear operators on V ).
4.2 Types of semigroups
Definition 4.2.1 (Contraction semigroups). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup with the prop-erty that
suptPr0,8q
StLpV q ď 1. (4.2)
Then S is called a contraction semigroup.
87
88 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
Definition 4.2.2 (Strongly continuous semigroups). Let K P tR,Cu, let pV, ¨V q bea normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup with the propertythat for all v P V it holds that the function
r0,8q Q t ÞÑ Stv P V (4.3)
is continuous. Then S is called strongly continuous.
Definition 4.2.3 (Uniformly continuous semigroups). Let K P tR,Cu, let pV, ¨V qbe a normed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup with theproperty that the function
r0,8q Q t ÞÑ St P LpV q (4.4)
is continuous. Then S is called uniformly continuous.
Example 4.2.4 (Matrix exponential). Let d P N and let A P Rdˆd be an arbitraryd ˆ d-matrix. Then the function r0,8q Q t ÞÑ eAt P Rdˆd is a uniformly continuoussemigroup.
Clearly, it holds that every uniformly continuous semigroup is also strongly con-tinuous. However, not every strongly continuous semigroup is uniformly continuoustoo. This is the subject of the next exercise.
Exercise 4.2.5. Give an example of a normed R-vector space pV, ¨V q and a stronglycontinuous semigroup S : r0,8q Ñ LpV q so that S is not a uniformly continuoussemigroup. Prove that your function S does indeed fulfill the desired properties.
4.3 The generator of a semigroup
Definition 4.3.1. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, andlet S : r0,8q Ñ LpV q be a semigroup. Then we denote by GS : DpGSq Ď V Ñ V thefunction with the property that
DpGSq “"
v P V : limtŒ0
„
Stv ´ v
t
exists
*
(4.5)
and with the property that for all v P DpGSq it holds that
GSv “ limtŒ0
„
Stv ´ v
t
(4.6)
and we call GS the (infinitesmal) generator of S.
4.3. THE GENERATOR OF A SEMIGROUP 89
Lemma 4.3.2 (Invariance of the domain of the generator). Let K P tR,Cu, letpV, ¨V q be a normed vector space, and let S : r0,8q Ñ LpV q be a semigroup. Thenit holds for all t P r0,8q that
St`
DpGSq˘
Ď DpGSq (4.7)
and it holds for all t P r0,8q, v P DpGSq that
GSStv “ StGSv. (4.8)
Proof of Lemma 4.3.2. Observe that for all t P r0,8q, v P DpGSq it holds that
limsŒ0
„
Ss rStvs ´ rStvs
s
“ limsŒ0
„
St
„
Ssv ´ v
s
“ St
„
limsŒ0
„
Ssv ´ v
s
“ StGSv. (4.9)
This completes the proof of Lemma 4.3.2.
In the next notion we label all linear operators that are generators of stronglycontinuous semigroups.
Definition 4.3.3 (Generator of a strongly continuous semigroup). Let K P tR,Cu,let pV, ¨V q be a K-Banach space, and let A : DpAq Ď V Ñ V be a linear operatorwith the property that there exists a strongly continuous semigroup S : r0,8q Ñ LpV qsuch that
GS “ A. (4.10)
Then we say that A is a generator of a strongly continuous semigroup.
We complete this section with a simple exercise which aims to illustrate and relatethe different concepts that have been introduced so far in this chapter.
Exercise 4.3.4. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space withV ‰ t0u, and let S : r0,8q Ñ LpV q be the function with the property that for allt P r0,8q it holds that
St “
#
IdV : t “ 0
0 : t ą 0. (4.11)
(i) Is S a semigroup? Prove that your answer is correct.
(ii) Is S a strongly continuous semigroup? Prove that your answer is correct.
(iii) Is S a uniformly continuous semigroup? Prove that your answer is correct.
(iv) Is S a contraction semigroup? Prove that your answer is correct.
(v) Specify DpGSq and GS.
90 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
4.4 A global a priori bound for semigroups
In the next result, Proposition 4.4.1, we present a global a priori bound for semigroupsof bounded linear operators.
Proposition 4.4.1 (A global a priori bound). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let S : r0,8q Ñ LpV q be a semigroup. Then it holds forall t P r0,8q, ε P p0,8q that
supsPr0,ts StLpV q ď“
supsPr0,εs SsLpV q‰
¨ et“
lnpSε1εLpV qq
‰`
. (4.12)
Proof of Proposition 4.4.1. Note that for all t P r0,8q, ε P p0,8q, n P N0Xptε´1, tεsit holds that
StLpV q “›
›Snε`pt´nεq›
›
LpV q“›
›SnεSpt´nεq›
›
LpV qď SnεLpV q
›
›Spt´nεq›
›
LpV q
“ rSεsnLpV q
›
›Spt´nεq›
›
LpV qď Sε
nLpV q
›
›Spt´nεq›
›
LpV q
ď“
supsPr0,εs SsLpV q‰
SεnLpV q ď
“
supsPr0,εs SsLpV q‰ “
max
1, SεLpV q(‰n
ď“
supsPr0,εs SsLpV q‰ “
max
1, SεLpV q(‰tε
““
supsPr0,εs SsLpV q‰
max!
e0, exp´
t ln´
Sε1εLpV q
¯¯)
ď“
supsPr0,εs SsLpV q‰
exp´
tmax
0, ln`
Sε1εLpV q
˘(
¯
.
(4.13)
This completes the proof of Proposition 4.4.1.
4.5 Strongly continuous semigroups
4.5.1 A priori bounds for strongly continuous semigroups
In Corollary 4.5.3 below we present a global priori bound for strongly continuoussemigroups. The proof of Corollary 4.5.3 uses the local a priori bound in Lemma 4.5.2below. The proof of Lemma 4.5.2 makes use of the uniform boundedness principle.This is the subject of the next result.
4.5. STRONGLY CONTINUOUS SEMIGROUPS 91
Theorem 4.5.1 (Uniform boundedness principle). Let K P tR,Cu, let pU, ¨Uq bea K-Banach space, let pV, ¨V q be a normed K-vector space, and let A Ď LpU, V q bea set with the property that for all u P U it holds that
supAPA
AuV ă 8. (4.14)
ThensupAPA
ALpU,V q ă 8. (4.15)
Lemma 4.5.2 (Local a priori bound). Let K P tR,Cu, let pV, ¨V q be a K-Banachspace, and let S : r0,8q Ñ LpV q be a semigroup which satisfies that for all v P V itholds that limtŒ0 Stv “ v. Then
lim suptŒ0
StLpV q “ limtŒ0
supsPr0,ts
SsLpV q ă 8. (4.16)
Proof of Lemma 4.5.2. We prove Lemma 4.5.2 by a contradiction. More specifically,we assume in the following that
limtŒ0
supsPr0,ts
SsLpV q “ 8. (4.17)
This and the fact that S0LpV q “ 1 ă 8 imply that for all t P p0,8q it holds that
supsPp0,ts
SsLpV q “ 8. (4.18)
Hence, there exists a strictly decreasing sequence tn P p0,8q, n P N, with limnÑ8 tn “0 and with the property that for all n P N it holds that
StnLpV q ě n. (4.19)
This ensures thatsupnPN
StnLpV q “ 8. (4.20)
Theorem 4.5.1 hence implies that there exists a vector v P V such that
supnPN
StnvV “ 8. (4.21)
Combining this and the fact that @n P N : StnvV ă 8 implies that
lim supnÑ8
StnvV “ 8. (4.22)
92 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
This and the assumption that @ v P V : limtŒ0 Stv “ v show that
8 ą vV “›
›
›limnÑ8
rStnvs›
›
›
V“ lim
nÑ8StnvV “ lim sup
nÑ8StnvV “ 8. (4.23)
This contradiction completes the proof of Lemma 4.5.2.
The next result, Corollary 4.5.3, proves a stronger version of Lemma 4.5.2. Ob-serve that Lemma 4.5.2 and Corollary 4.5.3 apply to strongly continuous semigroupson Banach spaces.
Corollary 4.5.3 (Global a priori bound). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a semigroup which satisfies that for allv P V it holds that limtŒ0 Stv “ v. Then it holds for all t P r0,8q, ε P p0,8q that
supsPr0,ts SsLpV q ď“
supsPr0,εs SsLpV q‰
¨ et“
lnpSε1εLpV qq
‰`
ă 8. (4.24)
Corollary 4.5.3 is an immediate consequence of Proposition 4.4.1 and Lemma 4.5.2above.
4.5.2 Pointwise convergence in the space of bounded linearoperators
Lemma 4.5.4 (A characterization of pointwise convergence in the space of boundedlinear operators). LetK P tR,Cu, let pV, ¨V q be aK-Banach space, and let pSnqnPN0 Ď
LpV q. Then @ v P V : limnÑ8 Snv ´ S0vV “ 0 if and only if for all compact setsK Ď V it holds that limnÑ8 supvPK Snv ´ S0vV “ 0.
Proof of Lemma 4.5.4. The proof of the “ð” direction in the statement of Lemma 4.5.4is clear. It thus remains to prove the “ñ” direction in the statement of Lemma 4.5.4.To this end we assume that for all v P V it holds that limnÑ8 Snv “ S0v and weassume that there exists a compact set K Ď V such that
lim supnÑ8
supvPK
Snv ´ S0vV ą 0. (4.25)
In the next step we note that there exists a sequence pvnqnPN Ď K such that for alln P N it holds that
Snvn ´ S0vnV “ supvPK
Snv ´ S0vV . (4.26)
4.5. STRONGLY CONTINUOUS SEMIGROUPS 93
The compactness of K ensures that there exist a w P K and a strictly increasingsequence pnkqkPN Ď N such that limkÑ8 vnk “ w. By assumption it holds thatlimkÑ8 Snkw “ S0w. This and Theorem 4.5.1 imply that
0 “ lim supkÑ8
Snkw ´ S0wV
“ lim supkÑ8
Snkpw ´ vnkq ` pSnk ´ S0q vnk ` S0pvnk ´ wqV
ě lim supkÑ8
pSnk ´ S0q vnkV ´ lim supkÑ8
Snkpw ´ vnkqV ´ limkÑ8
S0 pvnk ´ wqV
ě lim supkÑ8
supvPK
pSnk ´ S0q vV ´
„
supkPN
SnkLpV q
lim supkÑ8
w ´ vnkV
“ lim supkÑ8
supvPK
pSnk ´ S0q vV ą 0.
(4.27)
This condradiction completes the proof of Lemma 4.5.4.
4.5.3 Existence of solutions of linear ordinary differentialequations in Banach spaces
Lemma 4.5.5. Let K P tR,Cu, let pV, ¨V q be a K-Banach space, let S : r0,8q ÑLpV q be a strongly continuous semigroup, and let v P DpGSq. Then the functionr0,8q Q t ÞÑ Stv P V is continuously differentiable and it holds for all t P r0,8q that
ddtrStvs “ GSStv “ StGSv. (4.28)
Proof of Lemma 4.5.5. Observe that for all s, t P r0,8q with s ‰ t it holds that
›
›
›
›
Ssv ´ Stv
s´ t´ StGSv
›
›
›
›
V
“
›
›
›
›
Sminps,tq
„
Ss´minps,tqv ´ St´minps,tqv
s´ t
´ GSStv›
›
›
›
V
ď
›
›
›
›
Sminps,tq
„
Smaxps,tq´minps,tqv ´ v
maxps, tq ´minps, tq´ GSv
›
›
›
›
V
`›
›
“
Sminps,tq ´ St‰
GSv›
›
V
ď›
›Sminps,tq
›
›
LpV q
›
›
›
›
Smaxps,tq´minps,tqv ´ v
maxps, tq ´minps, tq´ GSv
›
›
›
›
V
`›
›
“
Sminps,tq ´ St‰
GSv›
›
V.
(4.29)
94 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
Corollary 4.5.3 and the fact that S is strongly continuous hence imply that for allt P r0,8q it holds that
limr0,8qQsÑt
›
›
›
›
Ssv ´ Stv
s´ t´ StGSv
›
›
›
›
V
ď
«
supsPr0,t`1s
SsLpV q
ff
„
limr0,8qQsÑt
›
›
›
›
Smaxps,tq´minps,tqv ´ v
maxps, tq ´minps, tq´ GSv
›
›
›
›
V
` limr0,8qQsÑt
›
›
“
Sminps,tq ´ St‰
GSv›
›
V“ 0.
(4.30)
This and Lemma 4.3.2 complete the proof of Lemma 4.5.5.
4.5.4 Domains of generators of strongly continuous semi-groups
In this subsection we prove that the generator of a strongly continuous semigroupis densily defined ; see Corollary 4.5.7 below. In the proof of Corollary 4.5.7 we usethe following result, Lemma 4.5.6. Lemma 4.5.6 and its proof can, e.g., be found asTheorem 2.4 (b) in Pazy [23] and Corollary 4.5.7 and its proof can, e.g., be found asCorollary 2.5 in Pazy [23].
Lemma 4.5.6 (Fundamental theorem of calculus for strongly continuous semi-groups). LetK P tR,Cu, t P r0,8q, let pV, ¨V q be aK-Banach space, let S : r0,8q Ñ
LpV q be a strongly continuous semigroup, and let v P V . Then it holds thatşt
0Ssv ds P
DpGSq and it holds that
GSˆż t
0
Ssv ds
˙
“ Stv ´ v. (4.31)
Proof of Lemma 4.5.6. Throughout this proof we assume w.l.o.g. that t P p0,8q.Then we observe that for all u P p0, tq it holds that
rSu ´ IdV s
u
„ż t
0
Ssv ds
“1
u
ż t
0
rSu`sv ´ Ssvs ds “1
u
ż t`u
t
Ssv ds´1
u
ż u
0
Ssv ds.
(4.32)Continuity of the function r0,8q Q s ÞÑ Ssv P V hence proves that
şt
0Ssv ds P DpGSq
and that
GSˆż t
0
Ssv ds
˙
“ limuŒ0
„
rSu ´ IdV s
u
„ż t
0
Ssv ds
“ Stv ´ S0v “ Stv ´ v. (4.33)
The proof of Lemma 4.5.6 is thus completed.
4.5. STRONGLY CONTINUOUS SEMIGROUPS 95
We are now ready to prove that the generator of a strongly continuous semigroupis densily defined.
Corollary 4.5.7. Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and letS : r0,8q Ñ LpV q be a strongly continuous semigroup. Then DpGSq is dense inV .
Proof of Corollary 4.5.7. Let v P V be arbitrary. The assumption that S is a stronglycontinuous semigroup together with the fundamental theorem of calculus ensures that
limtŒ0
ˆ
1
t
ż t
0
Ssv ds
˙
“ v. (4.34)
In addition, Lemma 4.5.6 proves that for all t P p0,8q it holds that 1t
şt
0Ssv ds P
DpGSq. This and (4.34) imply that v P DpGSq. The proof of Corollary 4.5.7 is thuscompleted.
4.5.5 Generators of strongly continuous semigroups
In this section we show that a strongly continuous semigroup is uniquely determinedby its generator; see Proposition 4.5.9 below. In Proposition 4.5.9 we use the assump-tion that the graph of one mapping is a subset of the graph of another mapping. Togetter a better understanding for this assumption, we first note the following remark.
Remark 4.5.8. Let A1, A2, B be sets and let f1 : A1 Ñ B and f2 : A2 Ñ B bemappings. Then it holds that Graphpf1q Ď Graphpf2q if and only if (A1 Ď A2 andf2|A1XA2 “ f1).
We are now ready to show that a strongly continuous semigroup is uniquelydetermined by its generator.
Proposition 4.5.9 (The generator determines the semigroup). Let K P tR,Cu, letpV, ¨V q be a K-Banach space, and let S, S : r0,8q Ñ LpV q be strongly continuoussemigroups with GraphpGSq Ď GraphpGSq. Then it holds that S “ S and GS “ GS.
Proof of Proposition 4.5.9. Let v P DpGSq Ď DpGSq, t P p0,8q and let η : r0, ts Ñ Vbe the function with the property that for all s P r0, ts it holds that
ηpsq “ St´s Ss v. (4.35)
96 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
Then it holds for all s P r0, ts, u P r0, ts with s ‰ u that
›
›
›
›
ηpuq ´ ηpsq
u´ s
›
›
›
›
V
“
›
›
›
›
›
St´u Su v ´ St´s Ss v
u´ s
›
›
›
›
›
V
“
›
›
›
›
›
St´u
„
Su v ´ Ss v
u´ s
`
“
St´u ´ St´s‰
Ssv
u´ s
›
›
›
›
›
V
“
›
›
›
›
›
St´s
„
Su v ´ Ss v
u´ s
`
”
St´u ´ St´s
ı
„
Su v ´ Ss v
u´ s
´
“
St´u ´ St´s‰
Ssv
pt´ uq ´ pt´ sq
›
›
›
›
›
V
ď
›
›
›
›
›
St´s
„
Su v ´ Ss v
u´ s
´
“
St´u ´ St´s‰
Ssv
pt´ uq ´ pt´ sq
›
›
›
›
›
V
`
›
›
›
›
”
St´u ´ St´s
ı
„
Su v ´ Ss v
u´ s
›
›
›
›
V
.
(4.36)
This implies that for all s P r0, ts, punqnPN Ď r0, tsztsu with limnÑ8 un “ s and alln P N it holds that
›
›
›
›
ηpunq ´ ηpsq
un ´ s
›
›
›
›
V
ď
›
›
›
›
›
St´s
„
Sunv ´ Ss v
un ´ s
´
“
St´un ´ St´s‰
Ssv
pt´ unq ´ pt´ sq
›
›
›
›
›
V
` sup!
›
›St´unw ´ St´sw›
›
V: w P tGSSsvu Y
!
Sumv´Ssvum´s
: m P N
))
.
(4.37)
Lemma 4.5.5 and Lemma 4.3.2 prove that for all s P r0, ts it holds that
limuÑs
«
“
St´u ´ St´s‰
Ssv
pt´ uq ´ pt´ sq
ff
“ GSSt´sSsv “ St´sGSSsv “ St´sGSSsv (4.38)
and
limuÑs
„
Suv ´ Ss v
u´ s
“ GSSsv. (4.39)
Putting (4.38)–(4.39) into (4.37) proves that for all s P r0, ts and all punqnPN Ď
4.5. STRONGLY CONTINUOUS SEMIGROUPS 97
r0, tsztsu with limnÑ8 un “ s it holds that
limnÑ8
›
›
›
›
ηpunq ´ ηpsq
un ´ s
›
›
›
›
V
ď limnÑ8
›
›
›
›
›
St´s
„
Sunv ´ Ss v
un ´ s
´
“
St´un ´ St´s‰
Ssv
pt´ unq ´ pt´ sq
›
›
›
›
›
V
` lim supnÑ8
sup!
›
›St´unw ´ St´sw›
›
V: w P tGSSsvu Y
!
Sumv´Ssvum´s
: m P N
))
“
›
›
›St´sGSSsv ´ St´sGSSsv
›
›
›
V
` lim supnÑ8
sup!
›
›St´unw ´ St´sw›
›
V: w P tGSSsvu Y
!
Sumv´Ssvum´s
: m P N
))
“ lim supnÑ8
sup!
›
›St´unw ´ St´sw›
›
V: w P tGSSsvu Y
!
Sumv´Ssvum´s
: m P N
))
.
(4.40)
This together with Lemma 4.5.5 and Lemma 4.5.4 proves that η is differentiable andthat for all s P r0, ts it holds that η1psq “ 0. This implies that
Stv “ ηp0q “ ηptq “ Stv. (4.41)
As v P DpGSq was arbitrary, we obtain that St|DpGSq “ St|DpGSq. Corollary 4.5.7 hence
proves that St “ St. This completes the proof of Proposition 4.5.9.
4.5.6 A generalization of matrix exponentials to infinite di-mensions
Proposition 4.5.9 and Definition 4.3.3 ensure that the next definition, Definition 4.5.10,makes sense.
Definition 4.5.10 (Generalized matrix exponential). LetK P tR,Cu, let pV, ¨V q bea K-Banach space, and let A : DpAq Ď V Ñ V be a generator of a strongly continuoussemigroup. Then we denote by eAt P LpV q, t P r0,8q, the linear operators with theproperty that for all strongly continuous semigroups S : r0,8q Ñ LpV q with GS “ Aand all t P r0,8q it holds that
eAt “ St. (4.42)
98 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
4.5.7 A characterization of strongly continuous semigroups
Lemma 4.5.11 (Characterization of strongly continuous semigroups). Let pV, ¨V qbe a Banach space. A semigroup S : r0,8q Ñ LpV q is strongly continuous if andonly if for all v P V it holds that limtŒ0 Stv “ v.
Proof of Lemma 4.5.11. A strongly continuous semigroup S : r0,8q Ñ LpV q clearlysatisfies that for all v P V it holds that limtŒ0 Stv “ v. In the following we thusassume that S : r0,8q Ñ LpV q is a semigroup which fulfills for all v P V thatlimtŒ0 Stv “ v. Corollary 4.5.3 hence implies that for all t P r0, T s it holds that
limsÑtSsv ´ StvV “ lim
sÑt
›
›Sminps,tq
`
S|t´s|v ´ v˘›
›
V
ď limsÑt
”
›
›Sminps,tq
›
›
LpV q
›
›S|t´s|v ´ v›
›
V
ı
ď
«
supuPr0,t`1s
SuLpV q
ff
”
limsÑt
›
›S|t´s|v ´ v›
›
V
ı
“ 0.
(4.43)
The proof of Lemma 4.5.11 is thus completed.
4.6 Uniformly continuous semigroups
Lemma 4.6.1. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and letS : r0,8q Ñ LpV q be a semigroup with the property that
limtŒ0St ´ S0LpV q “ 0. (4.44)
Then it holds for all t P r0,8q that supsPr0,ts SsLpV q ă 8.
Proof of Lemma 4.6.1. The assumption limtŒ0 St ´ S0LpV q “ 0 ensures that there
exists a real number ε P p0,8q such that
supsPr0,εs
SsLpV q ă 8. (4.45)
Combining this with Proposition 4.4.1 completes the proof of Lemma 4.6.1.
Lemma 4.6.2. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and letS : r0,8q Ñ LpV q be a semigroup. Then S is uniformly continuous if and only iflimtŒ0 St ´ S0LpV q “ 0.
4.6. UNIFORMLY CONTINUOUS SEMIGROUPS 99
Proof of Lemma 4.6.2. Clearly, it holds that if S is uniformly continuous, then itholds that limtŒ0 St ´ S0LpV q “ 0. It thus remains to prove that the condition
limtŒ0 St ´ S0LpV q “ 0 ensures that S is uniformly continuous. We thus assume inthe following that
limtŒ0St ´ S0LpV q “ 0. (4.46)
Lemma 4.6.1 hence implies that for all t P r0,8q it holds that
supsPr0,ts
SsLpV q ă 8. (4.47)
This and (4.46) show that for all t P r0,8q it holds that
limsÑtSs ´ StLpV q “ lim
sÑt
›
›Sminps,tq
“
Srmaxps,tq´minps,tqs ´ S0
‰›
›
LpV q
ď
”
limsÑt
›
›Srmaxps,tq´minps,tqs ´ S0
›
›
LpV q
ı
«
supsPr0,t`1s
SsLpV q
ff
“ 0.(4.48)
The proof of Lemma 4.6.2 is thus completed.
4.6.1 Matrix exponential in Banach spaces
Lemma 4.6.3. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, let A PLpV q, and let t P r0,8q. Then
8ÿ
n“0
›
›
›
›
pAtqn
n!
›
›
›
›
LpV q
ď
8ÿ
n“0
tn AnLpV qn!
“ etALpV q ă 8. (4.49)
The statement of Lemma 4.6.3 is clear. The next result, Lemma 4.6.4, demon-strates one way how uniformly continuous semigroup can be constructed. Observethat Lemma 4.6.3 ensures that the function S in Lemma 4.6.4 does indeed exist.
Lemma 4.6.4 (Matrix exponential in Banach spaces). Let K P tR,Cu, let pV, ¨V qbe a K-Banach space, let A P LpV q, and let S : r0,8q Ñ LpV q be the function withthe property that for all t P r0,8q it holds that
St “8ÿ
n“0
pAtqn
n!. (4.50)
Then it holds that S is a uniformly continuous semigroup, it holds that GS “ A, andit holds for all t P r0,8q that
StLpV q ď et GSLpV q . (4.51)
100 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
Proof of Lemma 4.6.4. First, we note that for all t1, t2 P r0,8q it holds that
St1St2 “
«
8ÿ
n“0
pAt1qn
n!
ff«
8ÿ
n“0
pAt2qn
n!
ff
“
8ÿ
n,m“0
An`m pt1qnpt2q
m
n!m!
“
8ÿ
k“0
ÿ
n,mPN0n`m“k
Ak pt1qnpt2q
m
n!m!“
8ÿ
k“0
Ak
k!
«
kÿ
n“0
k!
n! pk ´ nq!¨ pt1q
n¨ pt2q
pk´nq
ff
“
8ÿ
k“0
Ak
k!rt1 ` t2s
k“ St1`t2 .
(4.52)
This shows that S is a semigroup. Moreover, observe that for all t P r0,8q it holdsthat
St ´ S0LpV q “
›
›
›
›
›
8ÿ
n“1
pAtqn
n!
›
›
›
›
›
LpV q
ď t ALpV q
›
›
›
›
›
8ÿ
n“1
pAtqpn´1q
n!
›
›
›
›
›
LpV q
“ t ALpV q
›
›
›
›
›
8ÿ
n“0
pAtqn
pn` 1q!
›
›
›
›
›
LpV q
ď t ALpV q
«
8ÿ
n“0
AtnLpV qpn` 1q!
ff
ď t ALpV q etALpV q .
(4.53)
This proves that S is uniformly continuous. Furthermore, note that for all t P p0,8qit holds that
›
›
›
›
St ´ S0
t´ A
›
›
›
›
LpV q
“
›
›
›
›
›
A
«
8ÿ
n“1
pAtqpn´1q
n!
ff
´ A
›
›
›
›
›
LpV q
“
›
›
›
›
›
A
«
8ÿ
n“0
pAtqn
pn` 1q!
ff
´ A
›
›
›
›
›
LpV q
“
›
›
›
›
›
A
«
8ÿ
n“1
pAtqn
pn` 1q!
ff›
›
›
›
›
LpV q
“ t
›
›
›
›
›
A2
«
8ÿ
n“0
pAtqn
pn` 2q!
ff›
›
›
›
›
LpV q
ď t A2LpV q e
tALpV q .
(4.54)
This proves that GS “ A. Combining this with (4.49) completes the proof ofLemma 4.6.4.
4.6. UNIFORMLY CONTINUOUS SEMIGROUPS 101
4.6.2 Continuous invertibility of bounded linear operatorsin Banach spaces
Lemma 4.6.5 (Geometric series in Banach spaces and inversion of bounded linearoperators). Let K P tR,Cu, let pV, ¨V q be a K-Banach space, and let A P LpV q bea bounded linear operator with IdV ´ALpV q ă 1. Then it holds that A is bijective,
it holds that A´1 P LpV q, it holds thatř8
n“0 rIdV ´AsnLpV q ă 8, and it holds that
A´1“
8ÿ
n“0
rIdV ´Asn . (4.55)
Proof of Lemma 4.6.5. Throughout this proof let Q P LpV q and Sn P LpV q, n P N0,be the bounded linear operators with the property that for all n P N it holds that
Q “ IdV ´A and Sn “nÿ
k“0
Qk. (4.56)
Note that the assumption that QLpV q ă 1 ensures that
8ÿ
k“0
›
›Qk›
›
LpV qď
8ÿ
k“0
QkLpV q “1
“
1´ QLpV q‰ ă 8. (4.57)
This implies that Sn, n P N0, is a Cauchy-sequence in LpV q and thus convergence inLpV q. Next we claim that for all n P N0 it holds that
ASn “ IdV ´Qn`1. (4.58)
We show (4.58) by induction on n P N0. For this observe that
AS0 “ A IdV “ A “ IdV ´Q. (4.59)
This proves the base case n “ 0 in (4.58). Next note that if n P N and if ASn´1 “
IdV ´Qn, then it holds that
ASn “ A
«
nÿ
k“0
Qk
ff
“ A
«
IdV `nÿ
k“1
Qk
ff
“ A
«
IdV `Q
«
n´1ÿ
k“0
Qk
ffff
“ A rIdV `QSn´1s “ A`QASn´1 “ A`Q rIdV ´Qns
“ A`Q´Qn`1“ IdV ´Q
n`1.
(4.60)
102 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
This proves (4.58). Next note that (4.58) implies that for all n P N0 it holds that
ASn “ SnA “ IdV ´Qn`1. (4.61)
This and the fact that pSnqnPN0 Ď LpV q converges shows that
A”
limnÑ8
Sn
ı
loooomoooon
PLpV q
“
”
limnÑ8
Sn
ı
loooomoooon
PLpV q
A “ IdV . (4.62)
This implies that A is bijective and that A´1 “ limnÑ8 Sn P LpV q. The proof ofLemma 4.6.5 is thus completed.
4.6.3 Generators of uniformly continuous semigroup
Lemma 4.6.6 (The generator of a uniformly continuous semigroup). Let K P
tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a uniformlycontinuous semigroup. Then GS P LpV q.
Proof of Lemma 4.6.6. The assumption that S is uniformly continuous implies thatfor all t P r0,8q it holds that
limsŒ0
›
›
›
›
1
s
ż t`s
t
Su du´ St
›
›
›
›
LpV q
“ limsŒ0
›
›
›
›
1
s
ż t`s
t
rSu ´ Sts du
›
›
›
›
LpV q
ď limsŒ0
„
1
s
ż t`s
t
Su ´ StLpV q du
ď limsŒ0
«
supuPrt,t`ss
Su ´ StLpV q
ff
ď StLpV q
«
limsŒ0
«
supuPr0,ss
Su ´ S0LpV q
ffff
“ StLpV q
„
lim supsŒ0
Ss ´ S0LpV q
“ StLpV q
„
limsŒ0
Ss ´ S0LpV q
“ 0.
(4.63)
This implies that there exists a real number ε P p0,8q such that›
›
›
›
1
ε
ż ε
0
Ss ds´ S0
›
›
›
›
LpV q
ă 1. (4.64)
Lemma 4.6.5 hence shows thatşε
0Ss ds P LpV q is bijective with
„ż ε
0
Ss ds
´1
P LpV q. (4.65)
4.6. UNIFORMLY CONTINUOUS SEMIGROUPS 103
Hence, we obtain that for all t P p0, εq it holds that
St ´ IdVt
“
„
St ´ S0
t
„ż ε
0
Ss ds
„ż ε
0
Ss ds
´1
“
„
şε
0rSt`s ´ Sss ds
t
„ż ε
0
Ss ds
´1
“
«
şt`ε
tSs ds´
şε
0Ss ds
t
ff
„ż ε
0
Ss ds
´1
“
«
şε`t
εSs ds´
şt
0Ss ds
t
ff
„ż ε
0
Ss ds
´1
.
(4.66)
This together with the identity (4.63) shows that
limtŒ0
›
›
›
›
›
St ´ IdVt
´ rSε ´ S0s
„ż ε
0
Ss ds
´1›
›
›
›
›
LpV q
“ 0. (4.67)
This proves that GS P LpV q and that
GS “ rSε ´ S0s
„ż ε
0
Ss ds
´1
. (4.68)
The proof of Lemma 4.6.6 is thus completed.
4.6.4 A characterization result for uniformly continuous semi-groups
Theorem 4.6.7 (Characterization of uniformly continuous semigroups). Let K P
tR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a semigroup.Then the following statements are equivalent:
(i) It holds that S is uniformly continuous.
(ii) It holds that limtŒ0 St ´ S0LpV q “ 0.
(iii) It holds that GS P LpV q.
104 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
Proof of Theorem 4.6.7. Lemma 4.6.2 implies that (i) and (ii) are equivalent. More-over, Lemma 4.6.6 ensures that (i) implies (iii). It thus remains to prove that (iii)implies (i). To show this we assume for the rest of this proof that GS P LpV q. Thenlet S : r0,8q Ñ LpV q be the function with the property that for all t P r0,8q it holdsthat
St “8ÿ
n“0
ptGSqn
n!. (4.69)
Observe that Lemma 4.6.4 shows that S is uniformly continuous and that
GS “ GS. (4.70)
In the next step we apply Proposition 4.5.9 to obtain that S “ S. This proves thatS is uniformly continuous. The proof of Theorem 4.6.7 is thus completed.
4.6.5 An a priori bound for uniformly continuous semigroups
Combining Lemma 4.6.4 and Theorem 4.6.7 immediately results in the followingestimate.
Proposition 4.6.8 (A priori bounds for uniformly continuous semigroups). Let K PtR,Cu, let pV, ¨V q be a K-Banach space, and let S : r0,8q Ñ LpV q be a uniformlycontinuous semigroup. Then it holds for all t P r0,8q that
supsPr0,ts
SsLpV q ď et GSLpV q ă 8. (4.71)
4.7. SEMIGROUPS GENERATED BY DIAGONAL OPERATORS 105
4.7 Semigroups generated by diagonal operators
Theorem 4.7.1 (Semigroups generated by diagonal operators). Let K P tR,Cu, letpH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let B Ď H be an orthonormal basis, let λ : BÑK be a function with the property that supbPBRepλbq ă 8, and let A : DpAq Ď H Ñ
H be a linear operator with the property that
DpAq “
#
v P H :ÿ
bPB
|λb 〈b, v〉|2 ă 8
+
(4.72)
and with the property that for all v P DpAq it holds that Av “ř
bPB λb 〈b, v〉H b.Then it holds that A is a generator of a strongly continuous semigroup, it holds forall v P H, t P r0,8q that
eAtv “ÿ
bPB
eλbt 〈b, v〉H b, (4.73)
and it holds for all t P r0,8q that eAt P LpHq is a diagonal linear operator.
Proof of Theorem 4.7.1. Let S : r0,8q Ñ LpHq be the function with the propertythat for all v P H, t P r0,8q it holds that
Stpvq “ÿ
bPB
eλbt 〈b, v〉H b. (4.74)
Note that the assumption that supbPBRepλbq ă 8 ensures that such a function doesindeed exist. Next observe that for all t1, t2 P r0,8q, v P H it holds that
St1pSt2pvqq “ St1
˜
ÿ
bPB
eλbt2 〈b, v〉H b
¸
“ÿ
bPB
eλbt2 〈b, v〉H St1pbq
“ÿ
bPB
eλbt2 〈b, v〉H
«
ÿ
cPB
eλct1 〈c, b〉H c
ff
“ÿ
bPB
eλbt2 〈b, v〉H“
eλbt1b‰
“ÿ
bPB
eλbpt1`t2q 〈b, v〉H b “ St1`t2pvq.
(4.75)
The function S is thus a semigroup. Moreover, observe that Lebesgue’s theorem of
106 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
dominated convergence proves that for all v P H it holds that
limtŒ0Stv ´ v
2H “ lim
tŒ0
›
›
›
›
›
ÿ
bPB
“
eλbt ´ 1‰
〈b, v〉H b
›
›
›
›
›
2
H
“ limtŒ0
«
ÿ
bPB
›
›
“
eλbt ´ 1‰
〈b, v〉H b›
›
2
H
ff
“ limtŒ0
«
ÿ
bPB
ˇ
ˇeλbt ´ 1ˇ
ˇ
2|〈b, v〉H |
2
ff
“ limtŒ0
ż
B
ˇ
ˇeλbt ´ 1ˇ
ˇ
2|〈b, v〉H |
2 #Bpdbq
“
ż
B
limtŒ0
ˇ
ˇeλbt ´ 1ˇ
ˇ
2|〈b, v〉H |
2 #Bpdbq “ 0.
(4.76)
This completes the proof of Theorem 4.7.1.
Proposition 4.7.2 (Semigroups generated by diagonal operators). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A : DpAq Ď H Ñ H be a diagonallinear operator. Then it holds that supλPσP pAqRepλq ă 8 if and only if A is agenerator of a strongly continuous semigroup.
Proof of Proposition 4.7.2. Theorem 4.7.1 shows that the condition supλPσP pAqRepλq ă8 implies that A is a generator of a strongly continuous semigroup. In remainder ofthis proof we thus assume that A is the generator of a strongly continuous semigroupS : r0,8q Ñ LpHq. Then we note that the assumption that A is diagonal ensuresthat there exists an orthonormal basis B Ď H of H and a function λ : B Ñ K suchthat
DpAq “
#
v P H :ÿ
bPB
|λb 〈b, v〉H |2ă 8
+
(4.77)
and such that for all v P DpAq it holds that
Av “ÿ
bPB
λb 〈b, v〉H b. (4.78)
The fact that GS “ A implies that for all b P B it holds that the function
r0,8q Q t ÞÑ Stb P H (4.79)
is continuously differentiable and that for all t P r0,8q, v P H it holds that
〈v, S0pbq〉H “ 〈v, b〉H and ddt〈v, Stpbq〉H “ 〈v,GSStpbq〉H “ λb 〈v, Stpbq〉H .
(4.80)
4.7. SEMIGROUPS GENERATED BY DIAGONAL OPERATORS 107
This shows that for all b P B, v P H, t P r0,8q it holds that
〈v, Stb〉H “ eλbt 〈v, b〉H . (4.81)
Hence, we obtain that for all b P B, t P r0,8q it holds that
Stb “ eλbtb. (4.82)
This, in turn, ensures that
8 ą S1LpHq ě supbPB
S1bH “ supbPB
ˇ
ˇeλbˇ
ˇ “ supbPB
ˇ
ˇeRepλbqˇ
ˇ “ esupbPBRepλbq. (4.83)
This implies that supbPBRepλbq ă 8. The proof of Proposition 4.7.2 is thus com-pleted.
4.7.1 Semigroup generated by the Laplace operator
Example 4.7.3 (Heat equation with Dirichlet boundary conditions). Let A : DpAq ĎL2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace operator with Dirichlet bound-ary conditions on L2pBorelp0,1q; |¨|Rq and let v : p0, 1q Ñ R be a twice continuouslydifferentiable function with vp0`q “ vp1´q “ 0. Then it holds that supλPσP pAq λ “´π2 ă 8 and Theorem 4.7.1 hence ensures that A is the generator of a stronglycontinuous semigroup r0,8q Q t ÞÑ eAt P LpHq. Moreover, the function u : r0,8q ˆp0, 1q Ñ R with the property that
@ t P r0,8q, x P p0, 1q : upt, xq “ peAtvqpxq (4.84)
is twice continuously differentiable and satisfies that for all pt, xq P r0,8q ˆ p0, 1q itholds that
B
Btupt, xq “ B2
Bx2upt, xq, up0, xq “ vpxq, upt, 0`q “ upt, 1´q “ 0. (4.85)
108 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
Example 4.7.4 (Laplace operator on L2pBorelp0,1q; |¨|Rq without boundary condi-tions). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be a linear operatorwith DpAq “ H2pp0, 1q;Rq and with the property that for all v P DpAq it holds thatAv “ v2. Then A is not a diagonal linear operator in the sense of Definition 3.5.1 andA is not a generator of a strongly continuous semigroup. Indeed, observe that for allv P L2pBorelp0,1q; |¨|Rq with vpxq “ vpyq for all x, y P p0, 1q it holds that Av “ v2 “ 0and this shows that 0 P σP pAq and that 1 P Kernp0 ´ Aq “ KernpAq. Moreover,note that for all n P N, x P p0, 1q it holds that d2
dx2sinpnπxq “ ´n2π2 sinpnπxq and
this proves that for all n P N it holds that ´n2π2 P σP pAq and psinpnπxqqxPp0,1q P
Kernp´n2π2 ´ Aq. Furthemore, note thatş1
01 ¨ sinpπxq dx ‰ 0 and this implies that
it does not hold that for all v P Kernp0 ´ Aq, w P Kernp´n2π2 ´ Aq it holds thatş1
0vpxqwpxq dx “ 0. Proposition 3.5.13 hence proves that A is not a diagonal oper-
ator in the sense of Definition 3.5.1. Moreover, Proposition 4.5.9 implies that A isnot the generator of a strongly continuous semigroup.
4.7.2 Smoothing effect of the semigroup
Proposition 4.7.5. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, letA : DpAq Ď H Ñ H be a symmetric linear operator with infpσP pAqq ą 0, let B Ď Hbe an orthonormal basis, let λ : BÑ K be a function such that
DpAq “
#
v P H :ÿ
bPB
|λb 〈b, v〉H |2ă 8
+
(4.86)
and such that for all v P DpAq it holds that
Av “ÿ
bPB
λb 〈b, v〉H b, (4.87)
and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated toA. Then
• it holds that B Ď pXrPRHrq,
• it holds for all r P R that spanpBqHr“ Hr, and
• it holds for all r P R that A´rpBq “
bpλbqr
P H : b P B(
is an orthonormalbasis of Hr.
4.7. SEMIGROUPS GENERATED BY DIAGONAL OPERATORS 109
Proof of Proposition 4.7.5. Observe that Proposition 4.7.5 follows immediately fromDefinition 3.5.16, Definition 3.5.17, and Definition 3.5.25.
In the next result, Theorem 4.7.6, we establish a smoothing effect for strongly con-tinuous semigroups generated by diagonal linear operators. We recall Remark 2.5.21for the formulation of Theorem 4.7.6.
Theorem 4.7.6 (Smoothing effect of semigroups generated by diagonal operators).Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ Hbe a symmetric linear operator with suppσP pAqq ă 0, let B Ď H be an orthonormalbasis, let λ : BÑ K be a function such that
DpAq “
#
v P H :ÿ
bPB
|λb 〈b, v〉H |2ă 8
+
(4.88)
and such that for all v P DpAq it holds that
Av “ÿ
bPB
λb 〈b, v〉H b, (4.89)
and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to´A. Then
(i) it holds that for all r P r0,8q that
suptPr0,8q
›
›p´tAqreAt›
›
LpHqď
”r
e
ır
ă 8, (4.90)
(ii) it holds for all t P p0,8q, r P p´8, 0q, v P H that eAtvH ď |r||r| tr vHr ă 8,
(iii) it holds for all t P p0,8q, r P R that
eAtpHrq Ď pXsPRHsq (4.91)
(cf. Item (ii) and Proposition 2.5.19),
(iv) it holds for all t P r0,8q, r P p´8, 0q, v P H that etAvHr ď vHr , and
(v) it holds for all t P r0,8q, v P pYrPRHrq that
eAtv “ÿ
bPB
eλbt 〈b, v〉H b (4.92)
(cf. Item (iv) and Proposition 2.5.19).
110 CHAPTER 4. SEMIGROUPS OF BOUNDED LINEAR OPERATORS
Proof of Theorem 4.7.6. Observe that Proposition 4.7.5 implies that for all r P r0,8qit holds that
›
›p´tAqreAt›
›
LpHq“ sup
bPB
ˇ
ˇp´tλbqreλbt
ˇ
ˇ ď supxPp0,8q
„
xr
ex
ď
”r
e
ır
ă 8. (4.93)
The proof of Theorem 4.7.6 is thus completed.
Lemma 4.7.7. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and letA : DpAq Ď H Ñ H be a symmetric diagonal linear operator with suppσP pAqq ă 0.Then it holds for all t P p0,8q, r P r0, 1s that
›
›p´tAq´r`
eAt ´ IdH˘›
›
LpHqď 1. (4.94)
Proof of Lemma 4.7.7. Observe that for all t P p0,8q, r P r0, 1s it holds that
›
›p´tAq´r`
eAt ´ IdH˘›
›
LpHq“ sup
λPσP ptAqq
ˇ
ˇp´λq´r`
eλ ´ 1˘ˇ
ˇ
ď supxPp0,8q
„
p1´ e´xq
xr
ď 1.(4.95)
The proof of Lemma 4.7.7 is thus completed.
Exercise 4.7.8. Let T P p0,8q, r P r0, 1q, K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, let A : DpAq Ď H Ñ H be a symmetric diagonal linear operator withsup
`
σP pAq˘
ă 0, and let e : r0, T s Ñ H be a continuous function with the property
that for all t P r0, T s it holds that eptq “ ep0q `şt
0p´Aqr ept´sqA epsq ds. Prove that
suptPr0,T s eptqH ď ep0qH ¨ E1´r
“
T 1´r‰
.
Part II
Foundations in probability theory
111
Chapter 5
Random variables with values ininfinite dimensional spaces
In the most of this chapter we follow the presentations in Da Prato & Zabczyk [7],Werner [29] and Prevot & Rockner [24].
5.1 Borel sigma-algebras on normed vector spaces
5.1.1 The Hahn-Banach theorem
We first recall the Hahn-Banach theorem (see, e.g., Theorem III.1.5 in Werner [29]).
Theorem 5.1.1 (Hahn-Banach theorem; Extension of continuous linear functionals).Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, let U Ď V be a K-subspaceof V , and let φ P U 1. Then there exists a ϕ P V 1 with the property that
ϕ|U “ φ and ϕV 1 “ φU 1 . (5.1)
The proof of Theorem 5.1.1 uses the axiom of choice. The next corollary is animmediate consequence of the Hahn-Banach theorem.
Corollary 5.1.2 (Projections into 1-dimensional subspaces). Let K P tR,Cu, letpV, ¨V q be a normed K-vector space with V ‰ t0u, and let v P V . Then there existsa ϕ P V 1 with the property that
ϕpvq “ vV and ϕV 1 “ 1. (5.2)
113
114 CHAPTER 5. RANDOM VARIABLES
Proof of Corollary 5.1.2. We show Corollary 5.1.2 in two steps. In the first step weassume that v ‰ 0. Let U be the K-subspace of V given by U “ tλv P V : λ P Ku “spantvu and let φ : U Ñ K be the mapping with the property that for all λ P K itholds that
φpλvq “ λ vV . (5.3)
Theorem 5.1.1 implies the existence of a ϕ P V 1 with the property that
ϕ|U “ φ and ϕV 1 “ φU 1 “ 1. (5.4)
This proves (5.2) in the case v ‰ 0. In the second step we assume that v “ 0. Theassumption that V ‰ t0u then shows that there exists u P V with the property thatu ‰ 0. The first step hence proves that there exists a ϕ P V 1 with the property that
ϕpuq “ uV and ϕV 1 “ 1. (5.5)
In addition, observe that ϕpvq “ ϕp0q “ 0 “ vV . The proof of Corollary 5.1.2 isthus completed.
5.1.2 Norm representations in normed vector spaces
The next result, Corollary 5.1.3, is an immediate consequence of Corollary 5.1.2above.
Corollary 5.1.3 (Norm via the dual space). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space with V ‰ t0u, and let v P V . Then
vV “ supϕPV 1zt0u
ϕpvq
ϕV 1“ sup
ϕPV 1zt0u
|ϕpvq|
ϕV 1. (5.6)
If the normed vector space in Corollary 5.1.3 is separable, then the followingresult, Corollary 5.1.4, can be obtained. Corollary 5.1.4 is also an immediate conse-quence of Corollary 5.1.2 above.
Corollary 5.1.4 (Norm of a separable normed vector space via the dual space). LetK P tR,Cu and let pV, ¨V q be a separable normed K-vector space. Then there existsa sequence pϕnqnPN Ď V 1 with the property that for all v P V it holds that
vV “ supnPN
ϕnpvq “ supnPN
|ϕnpvq| . (5.7)
5.1. BOREL SIGMA-ALGEBRAS ON NORMED VECTOR SPACES 115
Proof of Corollary 5.1.4. W.l.o.g. we assume that V ‰ t0u. The assumption thatpV, ¨V q is separable implies that there exists a sequence vn P V , n P N, with theproperty that the set tvn : n P Nu is dense in V . Corollary 5.1.2 hence shows thatthere exists a sequence ϕn P V
1, n P N, with the property that for all n P N it holdsthat
ϕnpvnq “ vnV and ϕnV 1 “ 1. (5.8)
This implies that for all k P N it holds that
vkV “ supnPN
ϕnpvkq. (5.9)
Next let v P V and ε P p0,8q be arbitrary. Then observe that
supnPN
ϕnpvq ď supnPN
rϕnV 1 vV s “ vV . (5.10)
It thus remains to prove that
vV ď ε` supnPN
ϕnpvq. (5.11)
To see this observe that the fact that tvk P V : k P Nu is dense in V ensures thatthere exists a k P N such that v ´ vkV ď
ε2. This implies that
vV ď vkV ` v ´ vkV “ ϕkpvkq ` v ´ vkV“ ϕkpvq ` v ´ vkV ` ϕkpvk ´ vq
ď ϕkpvq ` v ´ vkV ` ϕkV 1 v ´ vkV“ ϕkpvq ` 2 v ´ vkV ď sup
nPNϕnpvq ` 2 v ´ vkV ď sup
nPNϕnpvq ` ε.
(5.12)
The proof of Corollary 5.1.4 is thus completed.
5.1.3 Linear characterization of the Borel sigma-algebra
Proposition 5.1.5 (Linear characterization of the Borel sigma-algebra). Let K P
tR,Cu and let pV, ¨V q be a separable normed K-vector space. Then there exists asequence ϕn P V
1, n P N, such that
BpV q “ σV ppϕqϕPV 1q “ σV pϕ : ϕ P V 1q “ σV pϕn : n P Nq . (5.13)
116 CHAPTER 5. RANDOM VARIABLES
Proof of Proposition 5.1.5. Let fv : V Ñ r0,8q, v P V , be the functions with theproperty that for all x, v P V it holds that
fvpxq “ x´ vV . (5.14)
Observe thatBpV q “ σV pfv : v P V q . (5.15)
Next observe that Corollary 5.1.4 implies that there exists a sequence ϕn P V1, n P N,
such that for all v P V it holds that
vV “ supnPN
ϕnpvq. (5.16)
This implies that
σV pϕn : n P Nq “ σV`
ϕnpp¨q ` vq : n P N, v P V˘
Ě σV pfv : v P V q . (5.17)
The proof of Proposition 5.1.5 is thus completed.
5.2 Measures on normed vector spaces
5.2.1 Uniqueness theorem for measures
Definition 5.2.1 (Image measure/Push forward measure). Let pΩ,A, µq be a mea-sure space, let pΩ, Aq be a measurable space, and let f : Ω Ñ Ω be an A/A-measurablemapping. Then we denote by fpµq “ fpµqA : AÑ r0,8s the function with the prop-erty that for all A P A it holds that
`
fpµq˘
pAq “`
fpµqA˘
pAq “ µ`
f´1pAq
˘
(5.18)
and we call fpµq (we call fpµqA) the image measure of f .
Definition 5.2.2 (Finiteness of measures). Let pΩ,F , µq be a measure space withµpΩq ă 8. Then µ is called finite.
Definition 5.2.3 (Sigma-finiteness of measures). Let pΩ,F , µq be a measure spacewith the property that there exists a sequence An P F , n P N, of sets such thatYnPNAn “ Ω and such that for all n P N it holds that µpAnq ă 8. Then µ is calledsigma-finite.
Definition 5.2.4 (X-Stability). Let Ω be a set and let E Ď PpΩq be a set with theproperty that for all a, b P E it holds that aX b P E. Then E is called X-stable.
5.2. MEASURES ON NORMED VECTOR SPACES 117
Theorem 5.2.5 (Uniqueness theorem for measures). Let Ω be a set, let E Ď PpΩqbe a X-stable subset of PpΩq, let µk : σΩpEq Ñ r0,8s, k P t1, 2u, be measures withthe property that µ1|E “ µ2|E and with the property that there exists a non-decreasingsequence Ωn P tA P E : µ1pAq ă 8u, n P N, such that YnPNΩn “ Ω. Then µ1 “ µ2.
5.2.2 Fourier transform of a measure
5.2.2.1 Characteristic functionals
Theorem 5.2.5 is, for example, established as Lemma 1.42 in Klenke [17].
Proposition 5.2.6 (Characteristic function). Let d P N and let µk : BpRdq Ñ r0,8s,k P t1, 2u, be finite measures with the property that for all ξ P Rd it holds that
ż
Rd
ei〈ξ,x〉Rd µ1pdxq “
ż
Rd
ei〈ξ,x〉Rd µ2pdxq. (5.19)
Then µ1 “ µ2.
Proposition 5.2.6 is, for example, proved as Theorem 15.8 in Klenke [18].
Definition 5.2.7 (Characteristic functional). Let pV, ¨V q be a normed R-vectorspace. Then we denote by
FV :
µ PMpBpV q, r0,8sq : µ is a finite measure on pV,BpV qq(
ÑMpV 1,Cq(5.20)
the mapping with the property that for all µ P DpFV q, ϕ P V1 it holds that
pFV µqpϕq “`
FV pµq˘
pϕq “
ż
V
ei¨ϕpxq µpdxq (5.21)
and for every µ P DpFV q we call FV pµq the characteristic functional of µ.
Lemma 5.2.8 (Elementary properties of the characteristic functionals). Let pV, ¨V qbe a normed R-vector space. Then
• for all µ P DpFV q it holds that pFV µqp0q “ µpV q,
• for all µ1, µ2 P DpFV q, a P r0,8q it holds that FV paµ1 ` µ2q “ aFV pµ1q `
FV pµ2q, and
• impFV q Ď CpV 1,Cq.
118 CHAPTER 5. RANDOM VARIABLES
Proof of Lemma 5.2.8. First of all, observe that for all µ P DpFV q it holds that
pFV µqp0q “
ż
V
ei0 µpdxq “
ż
V
1µpdxq “ µpV q. (5.22)
Next note that for all µ1, µ2 P DpFV q, a P r0,8q, ϕ P V1 it holds that
`
FV paµ1 ` µ2q˘
pϕq “
ż
V
eiϕpxq ra ¨ µ1pdxq ` µ2pdxqs
“ a
ż
V
eiϕpxqµ1pdxq `
ż
V
eiϕpxqµ2pdxq
“ aFV pµ1q ` FV pµ2q.
(5.23)
Finally, observe that Lebesgue’s theorem of dominated convergence proves that forall µ P DpFV q and all ϕn P V
1, n P N0, with limnÑ8 ϕn ´ ϕ0V 1 “ 0 it holds that
limnÑ8
|pFV µqpϕnq ´ pFV µqpϕ0q| ď limnÑ8
ż
V
ˇ
ˇeiϕnpxq ´ eiϕ0pxqˇ
ˇµpdxq
“
ż
V
limnÑ8
ˇ
ˇeiϕnpxq ´ eiϕ0pxqˇ
ˇµpdxq “ 0.
(5.24)
The proof of Lemma 5.2.8 is thus completed.
5.2.2.2 Fourier transform on separable normed vector spaces
Lemma 5.2.9 (Characteristic functional determines measure uniquely). Let pV, ¨V qbe a separable normed R-vector space. Then FV is injective.
Proof of Lemma 5.2.9. Let µ1, µ2 P DpFV q satisfy FV pµ1q “ FV pµ2q. Then notethat for all n P N, φ “ pφ1, . . . , φnq P LpV,R
nq, ξ P Rn it holds thatż
Rn
ei〈ξ,x〉Rn`
φpµ1q˘
pdxq “
ż
V
ei〈ξ,φpvq〉Rn pµ1qpdvq “`
FV µ1
˘
´
xξ, φp¨qyRn¯
“`
FV µ2
˘
´
xξ, φp¨qyRn¯
“
ż
V
ei〈ξ,φpvq〉Rn pµ2qpdvq “
ż
Rn
ei〈ξ,x〉Rn`
φpµ2q˘
pdxq.
(5.25)
Proposition 5.2.6 hence implies that for all n P N, φ P LpV,Rnq it holds that
φpµ1q “ φpµ2q. (5.26)
In the next step let E Ď PpV q be the set given by
E “ď
nPN
φ´1pBq P PpV q : φ P LpV,Rn
q, B P BpRnq(
. (5.27)
5.2. MEASURES ON NORMED VECTOR SPACES 119
Note that E Ď BpV q and observe that (5.26) shows that
µ1|E “ µ2|E . (5.28)
This, the fact that E is X-stable, the fact V P E , and Theorem 5.2.5 imply that
µ1|σV pEq “ µ2|σV pEq. (5.29)
Moreover, observe that Proposition 5.1.5 proves that
σV pEq “ BpV q. (5.30)
Combining this with (5.29) completes the proof of Lemma 5.2.9.
5.2.2.3 Almost surely separably supported
In Theorem 5.2.30 below we prove a generalization of Lemma 5.2.9. For this weneed a few preparations. These preparations and Theorem 5.2.30 are based on thepresentations in Van Neerven [27].
Lemma 5.2.10. Let pE, Eq be a topological space and let A Ď E be separable. ThenA is separable too.
Lemma 5.2.11. Let pE, Eq be a topological space and let A Ď E and B Ď E beseparable. Then AYB is separable too.
Lemma 5.2.12. Let pV, ¨V q be a normed vector space and let A Ď V be separable.Then spanpAq is separable too.
Definition 5.2.13 (Support of a measure). Let pE, Eq be a topological space and letµ : BpEq Ñ r0,8s be a measure on pE,BpEqq. Then we denote by supppµq the setgiven by
supppµq “ tx P E : p@U P E : x P U ñ µpUq ą 0qu (5.31)
and we call supppµq the support of µ.
Question 5.2.14. Let x P R. What is supp`
δRx |BpRq˘
?
Question 5.2.15. Let d P N. What is supppλRdq?
Exercise 5.2.16. Let pE, Eq be a topological space and let µ : BpEq Ñ r0,8s be ameasure on pE,BpEqq. Prove then that supppµq is a closed set in pE, Eq, i.e., provethat Ez supppµq P E.
120 CHAPTER 5. RANDOM VARIABLES
Remark 5.2.17. In general it is not true that µ`
Ez supppµq˘
“ 0. (see Wikipedia:support (measure theory)).
Definition 5.2.18 (Almost surely separably supported). Let pE, Eq be a topologicalspace and let µ : BpEq Ñ r0,8s be a measure with the property that there exists aseparable and closed subset A Ď E of E such that µpEzAq “ 0. Then µ is called a.s.separably supported (almost surely separably supported).
5.2.2.4 Trace set
Let pΩ,Fq be a measurable space (i.e., Ω is a set and F is a sigma-algebra on Ω)and let A Ď Ω be a subset of Ω. In some situations we are interested to have anappropriate measurable structure (an appropriate sigma-algebra) on A too. Thetrace set in the following definition provides an appropriate concept for this issue;see Lemma 5.2.20 below.
Definition 5.2.19 (Trace set). Let A and Ω be sets and let A Ď PpΩq be a subsetof the power set of Ω. Then we denote by A \A the set given by
A \A “ tAXB P PpAq : B P Au (5.32)
and we call A \A the trace set (of A in A).
Lemma 5.2.20 (Trace sigma-algebra). Let pΩ,Aq be a measurable space and letA Ď Ω be a subset of Ω (which is not necessarily an element of A). Then it holdsthat pA,A \Aq is a measurable space.
The proof of Lemma 5.2.20 is clear and therefore omitted. The next lemma andits proof can, e.g., be found as Corollary 1.83 in Klenke [18].
Lemma 5.2.21 (Trace sigma-algebras and generation of sigma-algebras). Let Ω bea set, let A Ď PpΩq be a subset of the power set of Ω, and let A Ď Ω be a subset ofΩ. Then
A \ σΩpAq “ σApA \Aq . (5.33)
Proof of Lemma 5.2.21. Let ι : A Ñ Ω be the mapping with the property that forall a P A it holds that ιpaq “ a. Then it holds for all B P PpΩq that
ι´1pBq “ ta P A : ιpaq P Bu “ AXB. (5.34)
5.2. MEASURES ON NORMED VECTOR SPACES 121
This implies that
σApA \Aq “ σAptAXB P PpAq : B P Auq “ σA`
ι´1pBq P PpAq : B P A
(˘
“
ι´1pBq P PpAq : B P σΩpAq
(
“ tAXB P PpAq : B P σΩpAqu “ A \ σΩpAq .(5.35)
The proof of Lemma 5.2.21 is thus completed.
Lemma 5.2.22 (Trace topology). Let pE, Eq be a topological space and let A Ď Ebe a subset of A. Then it holds that pA,A \ Eq is a topological space.
The proof of Lemma 5.2.22 is clear and therefore omitted.
Definition 5.2.23 (Generation of topologys). Let E be a set and let E Ď PpEq bea subset of the power set of E. Then we denote by τEpEq the set given by
τEpEq “č
A is a topologyon E with EĎA
A (5.36)
and we call τEpEq the smallest topology on E which contains E.
Lemma 5.2.24 (Topological spaces and continuous mappings). Let E and F be sets,let F Ď PpF q be a subset of the power set of F , and let f : E Ñ F be a mapping.Then
τE`
f´1pAq : A P F
(˘
“
f´1pAq : A P τF pFq
(
. (5.37)
Proof of Lemma 5.2.24. Throughout this proof let E Ď PpEq, E Ď PpEq and F ĎPpF q be the sets given by
E “ τE`
f´1pAq : A P F
(˘
, E “
f´1pAq : A P τF pFq
(
(5.38)
and F “
A P PpF q : f´1pAq P E
(
. (5.39)
Observe that pE, Eq, pE, Eq and pF, Fq are topological spaces and that F Ď F andtf´1pAq : A P Fu Ď E . Hence, we obtain that
τF pFq Ď F and E Ď E . (5.40)
This proves that
E “
f´1pAq : A P τF pFq
(
Ď
f´1pAq : A P F
(
Ď E Ď E . (5.41)
This shows that E “ E . The proof of Lemma 5.2.24 is thus completed.
122 CHAPTER 5. RANDOM VARIABLES
Lemma 5.2.25 (Trace topologys and generation of topologys). Let E be a set, letE Ď PpEq be a subset of the power set of E, and let A Ď E be a subset of E. Then
A \ τEpEq “ τApA \ Eq . (5.42)
Proof of Lemma 5.2.25. Let ι : A Ñ E be the mapping with the property that forall a P A it holds that ιpaq “ a. Then it holds for all B P PpΩq that
ι´1pBq “ ta P A : ιpaq P Bu “ AXB. (5.43)
This and Lemma 5.2.24 imply that
τApA \ Eq “ τAptAXB P PpAq : B P Euq “ τA`
ι´1pBq : B P E
(˘
“
ι´1pBq : B P τEpEq
(
“ tAXB P PpAq : B P τEpEqu “ A \ τEpEq .(5.44)
The proof of Lemma 5.2.22 is thus completed.
Lemma 5.2.26 (Open balls generate the topologys associated to a distance-typefunction). Let E be a set, let T Ď R be a set, and let d : E ˆ E Ñ T be a functionwith the property that @x, y, z P E : dpx, xq ď 0 and dpx, zq ď dpx, yq ` dpy, zq. Then
τpdq “ τE`
ty P E : dpx, yq ă εu P PpEq : x P E, ε P p0,8q(˘
. (5.45)
Lemma 5.2.26 is an immediate consequence from Defintion 2.5.3 and Lemma 2.5.4.In the next result, Corollary 5.2.27, we study the trace set of a topological space as-sociated to a metric.
Corollary 5.2.27 (Traces and metric spaces). Let pE, dEq be a metric space and letA Ď E be a subset of E. Then A \ τpdEq “ τpdE|AˆAq.
Proof of Corollary 5.2.27. Lemma 5.2.25 and Lemma 5.2.26 imply that
A \ τpdEq “ A \ τE`
ty P E : dEpx, yq ă εu P PpEq : x P E, ε P p0,8q(˘
“ τA`
A \
ty P E : dEpx, yq ă εu P PpEq : x P E, ε P p0,8q(˘
“ τA`
ty P A : dEpx, yq ă εu P PpAq : x P E, ε P p0,8q(˘
“ τA`
ty P A : dEpx, yq ă εu P PpAq : x P A, ε P p0,8q(˘
“ τpdE|AˆAq.
(5.46)
The proof of Corollary 5.2.27 is thus completed.
5.2. MEASURES ON NORMED VECTOR SPACES 123
Corollary 5.2.28 (Traces and Borel sigma algebras on topological spaces). LetpE, Eq be a topological space and let A Ď E be a subset of E. Then
A \ BpEq “ A \ σEpEq “ σApA \ Eq “ BpAq. (5.47)
Corollary 5.2.28 is an immediate consequence from Lemma 5.2.21. The nextresult, Corollary 5.2.29, specalises Corollary 5.2.28 to the case where the underlyingtopological space is generated by a metric.
Corollary 5.2.29 (Traces and Borel sigma algebras on metric spaces). Let pE, dEqbe a metric space and let A Ď E be a subset of E. Then
A \ BpEq “ A \ σE`
τpdEq˘
“ σA`
τpdE|AˆAq˘
“ BpAq. (5.48)
Corollary 5.2.29 is an immediate consequence of Corollary 5.2.28 and Corol-lary 5.2.27.
5.2.2.5 Fourier transform on normed vector spaces
In Lemma 5.2.9 it has been proved that the Fourier transforms of measures on sepa-rable normed vector spaces determine the measures uniquely. The next result, The-orem 5.2.30, provides a generalization to Lemma 5.2.9. The proof of Theorem 5.2.30uses Corollary 5.2.29 above.
Theorem 5.2.30 (Characteristic functional). Let pV, ¨V q be a normed R-vectorspace. Then FV |tµPDpFV q : µ is a.s. separably supportedu is injective.
Proof of Theorem 5.2.30. Let µ1, µ2 P DpFV q be two a.s. separably supported finitemeasures with the property that
FV pµ1q “ FV pµ2q (5.49)
The assumption that µ1 and µ2 are a.s. separably supported ensures that there existseparable and closed sets A1, A2 P BpV q with the property that
µ1pV zA1q “ µ2pV zA2q “ 0. (5.50)
Lemma 5.2.11 implies that the set A1 Y A2 is separable. This and Lemma 5.2.12prove that the set
spanpA1 Y A2q (5.51)
124 CHAPTER 5. RANDOM VARIABLES
is separable. Next let U be the set given by
U “ spanpA1 Y A2q. (5.52)
Lemma 5.2.10 and (5.50) prove that U is separable. The pair pU, ¨V |Uq is thus aclosed and separable R-vector subspace of pV, ¨V q. Moreover, equation (5.50) andthe fact that A1 Ď U and A2 Ď U prove that
µ1pV zUq “ µ2pV zUq “ 0. (5.53)
Next let ϕ P U 1 “ LpU,Rq be arbitrary. Theorem 5.1.1 then implies that there existsa ψ P V 1 “ LpV,Rq with the property that ψ|U “ ϕ. Equation (5.49) hence provesthat
`
FV µ1
˘
pψq “`
FV µ2
˘
pψq. (5.54)
Next note that Corollary 5.2.29 and the fact that U P BpV q imply that BpUq “U \BpV q “ PpUqXBpV q. This and (5.53) ensure that for all k P t1, 2u it holds that
`
FV µk˘
pψq “
ż
V
ei¨ψpxq µkpdxq “
ż
V
1Upxq ei¨ψpxq µkpdxq
“
ż
V
1Upxq ei¨ϕpxq µkpdxq “
ż
U
1Upxq ei¨ϕpxq µk|BpUqpdxq “
`
FU µk|BpUq˘
pϕq.
(5.55)
Combining (5.54) and (5.55) proves that`
FU µ1|BpUq˘
pϕq “`
FU µ2|BpUq˘
pϕq. Asϕ P U 1 was arbitrary, we obtain that
FU`
µ1|BpUq˘
“ FU`
µ2|BpUq˘
. (5.56)
Lemma 5.2.9 and the fact that U is separable hence imply that µ1|BpUq “ µ2|BpUq.This, (5.53) and the fact that BpUq “ U \ BpV q (see Corollary 5.2.29 above) implythat for all A P BpV q it holds that
µ1pAq “ µ1pAX Uq ` µ1pAzUq “ µ1pAX Uq “ µ1|U\BpV qpAX Uq
“ µ1|BpUqpAX Uq “ µ2|BpUqpAX Uq “ µ2pAX Uq
“ µ2pAzUq ` µ2pAX Uq “ µ2pAq.
(5.57)
The proof of Theorem 5.2.30 is thus completed.
Question 5.2.31. Let pV, ¨V q be a normed R-vector space and let µ : BpV q Ñ r0,8sbe a finite measure on pV,BpV qq. What is then the characteristic functional FV pµq?
Question 5.2.32. Let pV, ¨V q be a normed R-vector space and let µ1, µ2 : BpV q Ñr0,8s be two finite measures on pV,BpV qq with FV pµ1q “ FV pµ2q. Provide a condi-tion which is sufficient to ensure that µ1 “ µ2.
5.2. MEASURES ON NORMED VECTOR SPACES 125
5.2.3 Covariance of a measure
5.2.3.1 The Baire category theorem on complete metric spaces
Lemma 5.2.33 (A set contains an open ball). Let pE, dEq be a metric space andlet A Ď E. Then Ac ‰ E if and only if there exist ε P p0,8q, x P E such thatty P E : dEpx, yq ă εu Ď A.
Proof of Lemma 5.2.33. Observe that
Ac “ E
ô Ac is dense in E
ô @x P E : @ ε P p0,8q : D y P Ac : dEpx, yq ă ε
ô @x P E : @ ε P p0,8q : Ac X ty P E : dEpx, yq ă εu ‰ H.
(5.58)
This implies that
Ac ‰ E
ô Dx P E : D ε P p0,8q : Ac X ty P E : dEpx, yq ă εu “ H
ô Dx P E : D ε P p0,8q : ty P E : dEpx, yq ă εu Ď A.
(5.59)
The proof of Lemma 5.2.33 is thus completed.
Theorem 5.2.34 (Baire category theorem for complete metric spaces). Let pE, dEqbe a complete metric space and let An Ď E, n P N, be a sequence of closed subsetsof E with the property that rYnPNAns
c‰ E. Then there exists an N P N such that
rAN sc‰ E.
5.2.3.2 Regularities for correlations on normed vector spaces
Proposition 5.2.35 (A boundedness result for correlations on normed vector spaces).Let K P tR,Cu, r P p0,8q, let pV, ¨V q be a normed K-vector space, and letµ : BpV q Ñ r0,8s be a measure with the property that for all ϕ P V 1 it holds thatş
V|ϕpxq|r µpdxq ă 8. Then
supϕPV 1zt0u
„
ş
V|ϕpxq|r µpdxq
ϕrV 1
ă 8. (5.60)
126 CHAPTER 5. RANDOM VARIABLES
Proof of Proposition 5.2.35. Throughout this proof let Vn Ď V 1, n P N, be the setswith the property that for all n P N it holds that
Vn “"
ϕ P V 1 :
ż
V
|ϕpxq|r µpdxq ď n
*
. (5.61)
Fatou’s lemma proves that for all n P N, ψ P V 1, pϕkqkPN Ď Vn with limkÑ8 ϕk ´ ψV 1 “0 it holds that
ż
V
|ψpxq|r µpdxq “
ż
V
ˇ
ˇ
ˇlimkÑ8
ϕkpxqˇ
ˇ
ˇ
r
µpdxq “
ż
V
limkÑ8
|ϕkpxq|r µpdxq
“
ż
V
lim infkÑ8
|ϕkpxq|r µpdxq ď lim inf
kÑ8
ż
V
|ϕkpxq|r µpdxq ď lim inf
kÑ8rns “ n.
(5.62)
This implies that for every n P N it holds that Vn Ď V 1 is a closed subset of V 1. Theassumption that for all ϕ P V 1 it holds that
ş
V|ϕpxq|r µpdxq ă 8 proves that
YnPNVn “ V 1. (5.63)
The fact that Vn Ď V 1, n P N, are closed sets, the fact that pV 1, ¨V 1q is complete (seeLemma 3.4.10) and the Baire category theorem (see Theorem 5.2.34) hence prove
that there exists an N P N such that rVN sc ‰ V 1. Lemma 5.2.33 therefore showsthat there exist ψ P VN , ε P p0,8q such that
tϕ P V 1 : ϕ´ ψV 1 ď εu Ď VN . (5.64)
This implies that for all ϕ P V 1 with ϕV 1 ď ε it holds that
ż
V
|ϕpxq|r µpdxq “
ż
V
|pϕ` ψqpxq ´ ψpxq|r µpdxq
ď 2r„ż
V
|pϕ` ψqpxq|r µpdxq `
ż
V
|ψpxq|r µpdxq
ď 2r„
N `
ż
V
|ψpxq|r µpdxq
ď 2pr`1qN ă 8.
(5.65)
This, in turn, proves that for all ϕ P V 1zt0u it holds that
ż
V
|ϕpxq|r µpdxq “ϕrV 1
εr
ż
V
ˇ
ˇ
ˇ
ˇ
ε ¨ ϕpxq
ϕV 1
ˇ
ˇ
ˇ
ˇ
r
µpdxq ď2pr`1qN ϕrV 1
εră 8. (5.66)
This implies (5.60). The proof of Proposition 5.2.35 is thus completed.
5.2. MEASURES ON NORMED VECTOR SPACES 127
The next result, Corollary 5.2.36, specialises Proposition 5.2.35 to the case wherer P p0,8q in Proposition 5.2.35 is a natural number.
Corollary 5.2.36 (A continuity result for correlations on normed vector spaces).Let K P tR,Cu, k P N, let pV, ¨V q be a normed K-vector space, and let µ : BpV q Ñr0,8s be a measure with the property that for all ϕ P V 1 it holds that
ş
V|ϕpxq|k µpdxq ă
8. Then it holds that
supϕ1,...,ϕkPV 1zt0u
„
ş
V|ϕ1pxq ¨ . . . ¨ ϕkpxq|µpdxq
ϕ1V 1 ¨ . . . ¨ ϕkV 1
ă 8 (5.67)
and it holds that the symmetric k-linear form
V 1 ˆ ¨ ¨ ¨ ˆ V 1 Q pϕ1, . . . , ϕkq ÞÑ
ż
V
ϕ1pxq . . . ϕkpxqµpdxq P K (5.68)
is continuous.
Proof of Corollary 5.2.36. Proposition 5.2.35 implies that
supϕPV 1zt0u
«
ş
V|ϕpxq|k µpdxq
ϕkV 1
ff
ă 8. (5.69)
Holder’s inequality hence shows that
supϕ1,...,ϕkPV 1zt0u
„
ş
V|ϕ1pxq ¨ . . . ¨ ϕkpxq|µpdxq
ϕ1V 1 ¨ . . . ¨ ϕkV 1
ď supϕ1,...,ϕkPV 1zt0u
«
kź
l“1
«
ş
V|ϕlpxq|
k µpdxq
ϕlkV 1
ffff1k
“
«
supϕPV 1zt0u
ş
V|ϕpxq|k µpdxq
ϕkV 1
ff1k
ă 8.
(5.70)
This proves (5.67). Inequality (5.67), in turn, establishes (5.68). The proof of Corol-lary 5.2.36 is thus completed.
128 CHAPTER 5. RANDOM VARIABLES
5.2.3.3 Covariances of measures and random variables
Definition 5.2.37. Let K P tR,Cu, let pV, ¨V q be a normed K-vector space, andlet µ : BpV q Ñ r0,8s be a probability measure with the property that for all ϕ P V 1
it holds thatş
V|ϕpvq|2 µpdvq ă 8. Then we denote by Covpµq : V 1 ˆ V 1 Ñ K the
mapping with the property that for all ϕ, ψ P V 1 it holds that
pCov µqpϕ, ψq “`
Covpµq˘
pϕ, ψq
“
ż
V
„
ϕpvq ´
ż
V
ϕpuqµpduq
„
ψpvq ´
ż
V
ψpuqµpduq
µpdvq
“
ż
V
ϕpvqψpvqµpdvq ´
„ż
V
ϕpvqµpdvq
„ż
V
ψpvqµpdvq
(5.71)
and we call Covpµq the covariance of µ.
Definition 5.2.38 (Covariance of a random variable). Let pΩ,F ,Pq be a probabilityspace, let K P tR,Cu, let pV, ¨V q be a normed K-vector space, and let X : Ω Ñ Vbe an F/BpV q-measurable mapping with the property that for all ϕ P V 1 it holds thatE“
|ϕpXq|2‰
ă 8. Then we denote by CovpXq : V 1 ˆ V 1 Ñ K the mapping given byCovpXq “ CovpXpPqq.
Lemma 5.2.39 (Properties of the covariance). Let K P tR,Cu, let pV, ¨V q be anormed K-vector space, and let µ : BpV q Ñ r0,8s be a probability measure with theproperty that for all ϕ P V 1 it holds that
ş
V|ϕpvq|2 µpdvq ă 8. Then Covpµq : V 1 ˆ
V 1 Ñ K is
• nonnegative, i.e., @ϕ P V 1 : pCov µqpϕ, ϕq P r0,8q)
• Hermitian, i.e., @ϕ, ψ P V 1 : pCov µqpϕ, ψq “ pCov µqpψ, ϕq,
• sesquilinear, i.e., @φ, ϕ, ψ P V 1, a P K : pCov µqpaφ` ϕ, ψq “ apCov µqpφ, ψq `pCov µqpϕ, ψq,
• continuous, and
• it holds that
8 ą CovpµqLp2qpV 1,Kq “ supϕ,ψPV 1zt0u
„
|pCov µqpϕ, ψq|
ϕV 1 ψV 1
ď
ż
V
v2V µpdvq. (5.72)
5.2. MEASURES ON NORMED VECTOR SPACES 129
Proof of Lemma 5.2.39. First of all, observe that the nonnegativity, the Hermitian-ity, and the sesquilinearity of Covpµq follow immediately from Definition 5.2.37. Thecontinuity of Covpµq and the fact that
8 ą CovpµqLp2qpV 1,Rq “ supϕ,ψPV 1zt0u
„
|pCov µqpϕ, ψq|
ϕV 1 ψV 1
(5.73)
follow immediately from Corollary 5.2.36. In the next step we observe that Holder’sinequality implies that for all ϕ, ψ P V 1 it holds that
|pCov µqpϕ, ψq| ď
ż
V
ˇ
ˇ
ˇ
ˇ
ϕpvq ´
ż
V
ϕpuqµpduq
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ψpvq ´
ż
V
ψpuqµpduq
ˇ
ˇ
ˇ
ˇ
µpdvq
ď
«
ż
V
ˇ
ˇ
ˇ
ˇ
ϕpvq ´
ż
V
ϕpuqµpduq
ˇ
ˇ
ˇ
ˇ
2
µpdvq
ff12 «ż
V
ˇ
ˇ
ˇ
ˇ
ψpvq ´
ż
V
ψpuqµpduq
ˇ
ˇ
ˇ
ˇ
2
µpdvq
ff12
ď
„ż
V
|ϕpvq|2 µpdvq
12 „ż
V
|ψpvq|2 µpdvq
12
ď ϕV 1 ψV 1
ż
V
v2V µpdvq.
(5.74)
The proof of Lemma 5.2.39 is thus completed.
5.2.4 Gaussian measures on normed vector spaces
Definition 5.2.40 (One-dimensional Gaussian measures). Let µ : BpRq Ñ r0,8s bea measure with the property that there exist a, b P R such that for all B P BpRq itholds that
µpBq “
ż
txPR : ax`bPBu
1?
2πexp
ˆ
´y2
2
˙
dy. (5.75)
Then we call µ a one-dimensional Gaussian measure.
Definition 5.2.41 (Gaussian measures on possibly infinite dimensional spaces). LetpV, ¨V q be a normed R-vector space and let µ : BpV q Ñ r0,8s be a probability mea-sure with the property that for all ϕ P V 1 it holds that ϕpµqBpRq is a one-dimensionalGaussian measure. Then µ is called Gaussian (on pV, ¨V q).
130 CHAPTER 5. RANDOM VARIABLES
Example 5.2.42. Let T P p0,8q, m P N, let pΩ,F ,Pq be a probability space, letW : r0, T s ˆ Ω Ñ Rm be a standard Brownian motion with continous sample paths,and let W : Ω Ñ Cpr0, T s,Rmq be the mapping with the property that for all ω P Ω,t P r0, T s it holds that
`
W pωq˘
ptq “ Wtpωq. (5.76)
Then W pPq is a Gaussian measure on`
Cpr0, T s,Rmq, ¨Cpr0,T s,Rmq˘
. To see this
let ϕ P Cpr0, T s,Rmq1 be arbitrary and let PN : Cpr0, T s,Rmq Ñ Cpr0, T s,Rmq, N P
N, pN : Cpr0, T s,Rmq Ñ pRmqN`1, N P N, and ιN : pRmqN`1 Ñ Cpr0, T s,Rmq,N P N, be the mappings with the property that for all v P Cpr0, T s,Rmq, N P N,
pa0, a1, . . . , aNq P pRmqN`1, n P t0, 1, . . . , N ´ 1u, t P rnT
N, pn`1qT
Ns it holds that
`
PNpvq˘
ptq ““
n` 1´ tNT
‰
vpnTNq `
“
tNT´ n
‰
vp pn`1qTN
q, (5.77)
`
ιNpa0, a1, . . . , aNq˘
ptq ““
n` 1´ tNT
‰
an `“
tNT´ n
‰
an`1, (5.78)
pNpvq “`
vp0q, vp TNq, vp2T
Nq, . . . , vpT q
˘
. (5.79)
Observe that for all N P N it holds that PN , pN , and ιN are continuous and thatPN “ ιN ˝ pN . Next let ϕN : pRmqN`1 Ñ R, N P N, be the mappings with theproperty that for all N P N it holds that ϕN “ ϕ ˝ ιN . Moreover, note that for everyN P N it holds that
`
ϕ ˝ PN ˝ W˘
pPq “`
ϕ ˝ PN˘`
W pPq˘
“`
ϕN ˝ pN ˝ W˘
pPq (5.80)
is a Gaussian measure. This shows that for all y P R, N P N it holds that
E
”
exp´
i ¨ y ¨ ϕNppNpW qq¯ı
“ exp
ˆ
i ¨ y ¨E”
ϕNppNpW qqı
´y2
2¨ Var
´
ϕNppNpW qq¯
˙
“ exp
ˆ
´y2
2¨E
”
ˇ
ˇϕpPNpW qqˇ
ˇ
2ı
˙
. (5.81)
Observe that for all N P N, ω P Ω it holds that limNÑ8 ϕN`
pNpW pωqq˘
“ ϕpW pωqq.This, Lebesgue’s theorem of dominated convergence, and (5.81) imply that for ally P R it holds that
E
”
exp´
i ¨ y ¨ ϕpW q¯ı
“ limNÑ8
E
”
exp´
i ¨ y ¨ ϕNppNpW qq¯ı
“ limNÑ8
exp
ˆ
´y2
2¨E
”
ˇ
ˇϕpPNpW qqˇ
ˇ
2ı
˙
“ exp
ˆ
´y2
2¨ limNÑ8
E
”
ˇ
ˇϕpPNpW qqˇ
ˇ
2ı
˙
“ exp
ˆ
´y2
2¨E
”
ˇ
ˇϕpW qˇ
ˇ
2ı
˙
. (5.82)
This proves that ϕ`
W pP q˘
is a one-dimensional Gaussian measure and this shows
that W pPq is indeed a Gaussian measure.
5.2. MEASURES ON NORMED VECTOR SPACES 131
Lemma 5.2.43. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, and letµ : BpHq Ñ r0,8s be a measure. Then µ is Gaussian if and only if for all v P H itholds that Rep〈v, µ〉Hq is a Gaussian measure.
The proof of Lemma 5.2.43 is clear and therefore omitted.
5.2.4.1 Fourier transform of a Gaussian measure
Proposition 5.2.44 (Fourier transform of a Gaussian measure). Let pV, ¨V q be anormed R-vector space and let µ : BpV q Ñ r0,8s be a finite measure. Then µ isGaussian if and only if for all ϕ P V 1 it holds that
ş
V|ϕpvq|2 µpdvq ă 8 and
pFV µqpϕq “ exp
ˆ
i ∫Vϕpvqµpdvq ´ 1
2pCov µqpϕ, ϕq
˙
. (5.83)
Proof of Proposition 5.2.44. First of all, observe that if µ is a Gaussian measure,then it holds for all ϕ P V 1, ξ P R that
ş
V|ϕpvq|2 µpdvq ă 8 and
pFV µqpξ ¨ ϕq “
ż
V
ei¨ϕpvq¨ξ µpdvq “
ż
R
ei¨x¨ξ`
ϕpµq˘
pdxq
“ exp
ˆ
i ξ ∫Vϕpvqµpdvq ´ ξ2
2pCov µqpϕ, ϕq
˙
.
(5.84)
This proves the “ñ” direction in the statement of Proposition 5.2.44. It thus remainsto prove the “ð” direction in the statement of Proposition 5.2.44. To this end weassume in the following that for all ϕ P V 1 it holds that
ş
V|ϕpvq|2 µpdvq ă 8 and
pFV µqpϕq “ exp
ˆ
i ∫Vϕpvqµpdvq ´ 1
2pCov µqpϕ, ϕq
˙
. (5.85)
This implies that for all ϕ P V 1, ξ P R it holds that
pFV µqpξ ¨ ϕq “ exp
ˆ
i ξ ∫Vϕpvqµpdvq ´ ξ2
2pCov µqpϕ, ϕq
˙
. (5.86)
This, in turn, proves that for all ϕ P V 1 it holds that ϕpµq is a Gaussian measure onpR,BpRqq. The proof of Proposition 5.2.44 is thus completed.
Corollary 5.2.45 (Covariance of Gaussian measures). Let pV, ¨V q be a separablenormed R-vector space and let µk : BpV q Ñ r0,8s, k P t1, 2u, be Gaussian measureswith the property that Covpµ1q “ Covpµ2q and with the property that for all ϕ P V 1
it holds thatş
Vϕpvqµ1pdvq “
ş
Vϕpvqµ2pdvq. Then µ1 “ µ2.
Corollary 5.2.45 is an immediate consequence from Proposition 5.2.44 and fromLemma 5.2.9.
132 CHAPTER 5. RANDOM VARIABLES
5.3 Probability measures on Hilbert spaces
5.3.1 Nuclear operators on Hilbert spaces
Below we study Hilbert space valued random variables. The covariance operatorassociated to such a random variable is a nuclear operator. To study such covarianceoperators, we need a few more properties of nuclear operators.
Definition 5.3.1. Let K P tR,Cu and let V be a normed K-vector space. A mapping〈¨, ¨〉 : V ˆV Ñ K is called a scalar product/inner product on V if the following threeproperties are fulfilled:
(i) for all x P V zt0u it holds that 〈x, x〉 P p0,8q,
(ii) for all x, y P V it holds that 〈x, y〉 “ 〈y, x〉, and
(iii) for all λ P K, x, y, z P V it holds that 〈x, y ` λz〉 “ 〈x, y〉` λ 〈x, z〉.
Lemma 5.3.2 (Completeness of the space of nuclear operators). Let K P tR,Cu andlet pV, ¨V q and pW, ¨W q be K-Banach spaces. Then the pair pL1pV,W q, ¨L1pV,W q
q
is a K-Banach space.
Lemma 5.3.2 is, for example, proved as Theorem VI.5.3 (c) in Werner [29]. See,e.g., Lemma VI.5.6 in Werner [29] for the next lemma.
Lemma 5.3.3 (Operators with finite trace). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, let A P L1pHq, let B Ď H be an orthonormal basis of H, and letpvnqnPN Ď H, pwnqnPN Ď H satisfy that for all x P H it holds that
ř8
n“1 vnH wnH ă8 and
Ax “8ÿ
n“1
pwn b vnqpxq. (5.87)
Thenÿ
bPB
|〈b, Ab〉H | ă 8 and8ÿ
n“1
〈vn, wn〉H “ÿ
bPB
〈b, Ab〉H . (5.88)
5.3. PROBABILITY MEASURES ON HILBERT SPACES 133
Proof of Lemma 5.3.3. Observe that the Holder inequality proves that
ÿ
bPB
|〈b, Ab〉H | “ÿ
bPB
ˇ
ˇ
ˇ
ˇ
ˇ
⟨b,
8ÿ
n“1
pwn b vnqpbq
⟩H
ˇ
ˇ
ˇ
ˇ
ˇ
ďÿ
bPB
8ÿ
n“1
|〈b, pwn b vnqpbq〉H |
“ÿ
bPB
8ÿ
n“1
|〈b, wn〉H 〈vn, b〉H | ď8ÿ
n“1
«
ÿ
bPB
|〈b, wn〉H | |〈b, vn〉H |
ff
ď
8ÿ
n“1
«
ÿ
bPB
|〈b, wn〉H |2
ff12 «
ÿ
bPB
|〈b, vn〉H |2
ff12
“
8ÿ
n“1
vnH wnH ă 8.
(5.89)
Moreover, note that
8ÿ
n“1
〈vn, wn〉H “8ÿ
n“1
ÿ
bPB
〈b, vn〉H 〈b, wn〉H “8ÿ
n“1
ÿ
bPB
〈b, wn〉H 〈vn, b〉H
“
8ÿ
n“1
ÿ
bPB
〈b, pwn b vnqpbq〉H “ÿ
bPB
⟨b,
8ÿ
n“1
pwn b vnqpbq
⟩H
“ÿ
bPB
〈b, Ab〉H .(5.90)
The proof of Lemma 5.3.3 is thus completed.
Lemma 5.3.3 allow us to introduce the concept of the trace of a nuclear operatoron a Hilbert space.
Definition 5.3.4 (Trace of a nuclear operator). Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hqbe a K-Hilbert space, and let A P L1pHq. Then we denote by traceHpAq P K theelement of K with the property that for all orthonormal basis B Ď H it holds that
traceHpAq “ÿ
bPB
〈b, Ab〉H P K. (5.91)
Lemma 5.3.5. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be K-Hilbert spaces, and let A P LpH,Uq. Then it holds that A˚A P LpHq is nonnegativeand symmetric.
Proof of Lemma 5.3.5. Note that for all v, w P H it holds that
〈v, A˚Aw〉H “⟨rA˚s˚ v,Aw
⟩U“ 〈Av,Aw〉U “ 〈A
˚Av,w〉H . (5.92)
This proves that A˚A is symmetric and that for all v P H it holds that
〈v, A˚Av〉H “ 〈Av,Av〉U “ Av2U ě 0. (5.93)
Hence, we obtain that A˚A is nonnegative. The proof of Lemma 5.3.5 is thus com-pleted.
134 CHAPTER 5. RANDOM VARIABLES
Lemma 5.3.5 and Definiton 3.4.13 allows us to introduce the absolute value op-erator of a bounded linear operator.
Definition 5.3.6 (The absolute value operator of a bounded linear operator). LetK P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be K-Hilbert spaces, and letA P LpH,Uq. Then we denote by |A| P LpHq the linear operator given by
|A| “ rA˚As12P LpHq. (5.94)
Lemma 5.3.7 (The trace of the absolute value operator). Let K P tR,Cu, letpV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-Hilbert spaces, let A P LpV,W q, and letB Ď V be an orthonormal basis of V . Then
ř
bPB 〈b, |A| b〉V ă 8 if and only ifA P L1pV,W q and in that case it holds that
ÿ
bPB
〈b, |A| b〉V “ traceV p|A|q “ AL1pV,W q. (5.95)
Lemma 5.3.7 can, e.g., be established by using the theory of singular values; see,e.g., Werner [29].
Proposition 5.3.8 (Properties of the absolute value operator of a bounded linearoperator). Let K P tR,Cu, let pV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-Hilbertspaces, and let A P LpV,W q. Then
(i) |A| P LpV q is nonnegative and symmetric,
(ii)ˇ
ˇ |A|ˇ
ˇ “ |A|,
(iii) for all v P V it holds that
AvW “ |A|vV , (5.96)
(iv) for all i P t1, 2u it holds that A P LipV,W q if and only if |A| P LipV q,
(v) A P L2pV,W q if and only if |A|2 “ A˚A P L1pV q and it that case it holds that
A2L2pV,W q“›
›|A|2›
›
L1pV q“ traceV p|A|
2q, (5.97)
5.3. PROBABILITY MEASURES ON HILBERT SPACES 135
(vi) A P L1pV,W q if and only if |A|12P L2pV q and it that case it holds that
AL1pV,W q“›
›|A|12›
›
2
L2pV q“ traceV p|A|q, (5.98)
and
(vii) if pV, 〈¨, ¨〉V , ¨V q “ pW, 〈¨, ¨〉W , ¨W q and if A is symmetric and nonnegative,then |A| “ A.
Proof of Proposition 5.3.8. Definition 3.4.13 ensures that |A| is nonnegative and
symmetric. This shows that ||A|| ““
|A|˚ |A|‰12
““
|A|2‰12
“ |A|. Furthermore,observe that for all v P V it holds that
Av2W “ 〈Av,Av〉W “ 〈v, A˚Av〉V “@
v, |A| |A| vD
V
“@
|A| v, |A| vD
V“ |A|v2V .
(5.99)
This proves (5.96). The identity (5.96), in turn, shows that A P L2pV,W q if and onlyif |A| P L2pV q. Lemma 5.3.7 implies that A P L1pV,W q if and only if there exists anorthonormal basis B Ď V of V such that
ÿ
bPB
〈b, |A| b〉V ă 8. (5.100)
Furthermore, Lemma 5.3.7 proves that |A| P L1pV q if and only if there exists anorthonormal basis B Ď V of V such that
ÿ
bPB
〈b, |A| b〉V ă 8. (5.101)
Combining (5.100) and (5.101) proves that A P L1pV,W q if and only if |A| P L1pV q.Next let B Ď V be an orthonormal basis of V and observe that
ÿ
bPB
Ab2W “ÿ
bPB
〈Ab,Ab〉V “ÿ
bPB
〈b, A˚Ab〉V “ÿ
bPB
⟨b, |A|2 b
⟩V. (5.102)
This and Lemma 5.3.7 prove Item (v). Item (iv), Item (v) and Lemma 5.3.7 proveItem (vi). Moreover, note that if pV, 〈¨, ¨〉V , ¨V q “ pW, 〈¨, ¨〉W , ¨W q and if A issymmetric and nonnegative, then
rA˚As12““
A2‰12“ A. (5.103)
This completes the proof of Proposition 5.3.8.
136 CHAPTER 5. RANDOM VARIABLES
Definition 5.3.9 (Rank-1 operators in Hilbert spaces; cf. Definition 3.4.19). LetK P tR,Cu, let pV, 〈¨, ¨〉V , ¨V q and pW, 〈¨, ¨〉W , ¨W q be K-Hilbert spaces, and letv P V , w P W . Then we denote by w b v P LpV,W q the linear operator with theproperty that for all u P V it holds that
pw b vqpuq “`
w b 〈v, ¨〉V˘
puq “ w 〈v, u〉V . (5.104)
Note, in the setting of Definition 5.3.9, that
w b v P L1pV,W q Ď L2pV,W q “ HSpV,W q Ď LpV,W q (5.105)
and
w b vL1pV,W q“ w b vL2pV,W q
“ w b vLpV,W q “ wW vV . (5.106)
Theorem 5.3.10 (Spectral decomposition for compact operators). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let A P KpH,Hq be a symmetric com-pact operator. Then A is a diagonal linear operator in the sense of Definition 3.5.1.
Theorem 5.3.10 is, e.g., proved as Theorem VI.3.2 in Werner [29].
5.3.2 Expectation and covariance operator
Definition 5.3.11 (Covariance operator of a probability measure on a Hilbertspace). Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let µ : BpHq Ñ r0,8s be aprobability measure with the property that for all w P H it holds that
ş
H|〈w, v〉H |
2 µpdvq ă8. Then we denote by CovOppµq P LpHq the unique bounded linear operator suchthat for all v, w P H it holds that
〈v,CovOppµqw〉H “ pCov µq`
〈v, ¨〉H , 〈w, ¨〉H˘
. (5.107)
Lemma 5.3.12 (Properites of covariance operators). Let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let µ : BpHq Ñ r0,8s be a probability measure with
ş
Hv2H µpdvq ă
8. Then CovOppµq is a symmetric and nonnegative nuclear operator and it holdsthat
traceHpCovOppµqq “ CovOppµqL1pHqď
ż
H
v2H µpdvq P r0,8q. (5.108)
5.3. PROBABILITY MEASURES ON HILBERT SPACES 137
Proof of Lemma 5.3.12. Nonnegativity and symmetry of CovOppµq follows immedi-ately from Lemma 5.2.39. Next observe that for all orthonormal bases B Ď H of Hit holds thatÿ
bPB
〈b, |CovOppµq| b〉H “ÿ
bPB
〈b,CovOppµqb〉H
“ÿ
bPB
«
ż
H
|〈b, v〉H |2 µpdvq ´
ˇ
ˇ
ˇ
ˇ
ż
H
〈b, v〉H µpdvqˇ
ˇ
ˇ
ˇ
2ff
ďÿ
bPB
ż
H
|〈b, v〉H |2 µpdvq “ sup
BĎBfinite
ÿ
bPB
ż
H
|〈b, v〉H |2 µpdvq
“ supBĎBfinite
ż
H
ÿ
bPB
|〈b, v〉H |2 µpdvq ď sup
BĎBfinite
ż
H
v2H µpdvq
“
ż
H
v2H µpdvq.
(5.109)
Lemma 5.3.7 hence completes the proof of Lemma 5.3.12.
Definition 5.3.13 (Covariance operator of a Hilbert space valued random variable).Let pΩ,F ,Pq be a probability space, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, and letX : Ω Ñ H be an F/BpHq-measurable mapping with the property that for all v P Hit holds that E
“
|〈v,X〉H |2‰
ă 8. Then we denote by CovOppXq P LpHq the linearoperator given by CovOppXq “ CovOp
`
XpPq˘
.
Note, in the setting of Definition 5.3.11, that for all v, w P H it holds that
〈v,CovOppXqw〉H “ E”
`
〈v,X〉H´Er〈v,X〉Hs˘`
〈w,X〉H´Er〈w,X〉Hs˘
ı
. (5.110)
Definition 5.3.14. Let pΩ,F ,Pq be a probability space, let pV, ¨V q be a Banachspace, and let X P L1pP; ¨V q. Then we denote by ErXs P V the element from Vgiven by
ErXs “
ż
Ω
XpωqPpdωq. (5.111)
Note that the integral appearing on the right hand side of (5.111) is a Bochnerintegral; see Section 3.6 for details.
138 CHAPTER 5. RANDOM VARIABLES
Proposition 5.3.15 (Properties of the covariance operator of a Hilbert space valuedrandom variable). Let pΩ,F ,Pq be a probability space, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, and let X P L2pP; ¨Hq. Then
(i) it holds that CovOppXq is a symmetric and nonnegative nuclear operator,
(ii) it holds that
CovOppXq “ E“`
X ´ErXs˘
b`
X ´ErXs˘‰
P L1pHq, (5.112)
and
(iii) it holds that
traceHpCovOppXqq “ E“
X ´ErXs2H‰
“ CovOppXqL1pHqP r0,8q.
(5.113)
5.3.3 Karhunen-Loeve expansion
Theorem 5.3.16 (Karhunen-Loeve expansion). Let pΩ,F ,Pq be a probability space,let pH, 〈¨, ¨〉H , ¨Hq be a separable R-Hilbert space, let X P L2pP; ¨Hq, let B Ď Hbe an orthonormal basis of H, and let λ : BÑ r0,8q be a globally bounded functionsuch that for all v P H it holds that CovOppXq v “
ř
bPB λb 〈b, v〉H b. Then
(i) the random variables p〈b,X ´ErXs〉HqbPB are centered and pairwise uncorre-lated,
(ii) it holds for all b P B that Varp〈b,X ´ErXs〉Hq “ Varp〈b,X〉Hq “ λb,
(iii) it holds that
X “ ErXs `ÿ
bPB
〈b,X ´ErXs〉H b, (5.114)
and
(iv) it holds for all B Ď B that›
›
›
›
›
X ´
«
ErXs `ÿ
bPB
〈b,X ´ErXs〉H b
ff›
›
›
›
›
L2pP;¨Hq
“
d
ÿ
bPBzB
λb ă 8. (5.115)
5.3. PROBABILITY MEASURES ON HILBERT SPACES 139
Proof of Theorem 5.3.16. Equation (5.114) follows immediately from the fact thatB is an orthonormal basis of H. Furthermore, note that the random variablesp〈b,X ´ErXs〉HqbPB are centered. Next observe that for all b1, b2 P B it holds that
Er〈b1, X ´ErXs〉H 〈b2, X ´ErXs〉Hs “ 〈b1,CovOppXqb2〉H“
ÿ
bPB
〈b1, λb 〈b, b2〉H b〉H
“ λb1 〈b1, b2〉H .
(5.116)
This implies that the random variables p〈b,X ´ErXs〉HqbPB are pairwise uncorre-lated and it shows that for all b P B it holds that
Varp〈b,X〉Hq “ Varp〈b,X ´ErXs〉Hq “ λb. (5.117)
Combining this with (5.114) proves that for all B Ď B it holds that
›
›
›
›
›
X ´
«
ErXs `ÿ
bPB
〈b,X ´ErXs〉H b
ff›
›
›
›
›
L2pP;¨Hq
“
›
›
›
›
›
›
ÿ
bPBzB
〈b,X ´ErXs〉H b
›
›
›
›
›
›
L2pP;¨Hq
“
d
ÿ
bPBzB
E“
|〈b,X ´ErXs〉H |2‰
“
d
ÿ
bPBzB
λb.
(5.118)
The proof of Theorem 5.3.16 is thus completed.
5.3.4 Gaussian measures on Hilbert spaces
5.3.4.1 Karhunen-Loeve expansion
Definition 5.3.17 (Gaussian distributed random variables). Let pΩ,F ,Pq be a prob-ability space, let pV, ¨V q be a normed R-vector space, and let X : Ω Ñ V be anF/BpV q-measurable mapping with the property that XpPq is a Gaussian measure.Then X is called Gaussian distributed.
140 CHAPTER 5. RANDOM VARIABLES
Corollary 5.3.18 (Karhunen-Loeve expansion for Gaussian distributed randomvariables). Let pΩ,F ,Pq be a probability space, let pH, 〈¨, ¨〉H , ¨Hq be a separableR-Hilbert space, let X P L2pP; ¨Hq be Gaussian distributed, let B Ď H be an or-thonormal basis of H, and let λ : BÑ r0,8q be a globally bounded function such thatfor all v P H it holds that CovOppXq v “
ř
bPB λb 〈b, v〉H b. Then the random vari-ables 1?
λb〈b,X ´ErXs〉H , b P λ´1pp0,8qq, are independent identically distributed
(i.i.d.) standard normal random variables and it holds P-a.s. that
X “ ErXs `ÿ
bPλ´1pp0,8qq
a
λb
„
〈b,X ´ErXs〉H?λb
b. (5.119)
Proof of Corollary 5.3.18. First of all, observe that Theorem 5.3.16 together withthe assumption that XpPq is a Gaussian measure imply that the random variables
1?λb〈b,X ´ErXs〉H , b P λ´1pp0,8qq, are i.i.d. standard normal random variables.
Moreover, Theorem 5.3.16 ensures that for all b P λ´1pt0uq it holds that
E“
|〈b,X ´ErXs〉H |2‰
“ Varp〈b,X ´ErXs〉Hq “ λb “ 0. (5.120)
This proves that for all b P λ´1pt0uq it holds P-a.s. that
〈b,X ´ErXs〉H “ 0. (5.121)
This shows that it holds P-a.s. that
X ´ErXs “ÿ
bPλ´1pp0,8qq
〈b,X ´ErXs〉H b. (5.122)
Equation (5.122) implies (5.119). The proof of Corollary 5.3.18 is thus completed.
5.3.4.2 Construction of Gaussian measures on Hilbert spaces
In Theorem 5.3.21 below we establish the existence of Gaussian measures on Hilbertspaces. In the proof of Theorem 5.3.21 we use the fact that the set of Cauchysequences of a sequence of strongly measurable mappings is a measurable set; seeLemma 5.3.19 below.
Lemma 5.3.19 (Cauchy sequence). Let pΩ,Fq be a measurable space, let pE, dEqbe a metric space, and let Xn : Ω Ñ E, n P N, be F/pE, dEq-strongly measurablemappings. Then it holds that
ω P Ω: pXnpωqqnPN is a Cauchy-sequence(
P F . (5.123)
5.3. PROBABILITY MEASURES ON HILBERT SPACES 141
Proof of Lemma 5.3.19. First of all, note that the assumption that Xn, n P N,are F/pE, dEq-strongly measurable ensures that for all n,m P N it holds thatpXn, Xmq : Ω Ñ E ˆ E is F/BpE ˆ Eq-measurable. The continuity of the map-ping dE : E ˆ E Ñ r0,8q hence implies that for all n,m P N it holds that thefunction
Ω Q ω ÞÑ dEpXnpωq, Xmpωqq P r0,8q (5.124)
is F/Bpr0,8qq-measurable. This ensures that for all k, n,m P N it holds that
tdEpXn, Xmq ă 1ku P F . (5.125)
This implies that
ω P Ω: pXnpωqqnPN is a Cauchy-sequence(
“
ω P Ω: @ ε P p0,8q : DN P N : @n,m P NX rN,8q : dEpXnpωq, Xmpωqq ă ε(
“
ω P Ω: @ k P N : DN P N : @n,m P NX rN,8q : dEpXnpωq, Xmpωqq ă 1k(
“ XkPN YNPN Xn,mPNXrN,8q tω P Ω: dEpXnpωq, Xmpωqq ă 1ku
“ XkPN YNPN Xn,mPtN,N`1,... u tdEpXn, Xmq ă 1kulooooooooooomooooooooooon
PF
P F .
(5.126)
This completes the proof of Lemma 5.3.19.
The next result, Corollary 5.3.20, is a direct consequence of Proposition 5.2.44above.
Corollary 5.3.20 (Fourier transform of a Gaussian measure on a Hilbert space). LetpH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space and let µ : BpHq Ñ r0,8s be a finite measure.Then µ is Gaussian if and only if for all v P H it holds that
ş
H|〈v, w〉H |
2 µpdwq ă 8and
pFV µqp〈v, ¨〉Hq “ exp`
i ∫H 〈v, w〉H µpdwq ´12〈v,CovOppµq v〉H
˘
. (5.127)
The next result, Theorem 5.3.21, establishes the existence of a Gaussian measureon a Hilbert space with a given mean vector and a given nuclear covariance operator.
Theorem 5.3.21 (Gaussian measures on Hilbert spaces). Let pH, 〈¨, ¨〉H , ¨Hq bean R-Hilbert space, let v P H, and let Q P L1pHq be a nonnegative and symmetricnuclear operator. Then there exists a Gaussian measure Nv,Q : BpHq Ñ r0,8s withthe property that CovOppNv,Qq “ Q and with the property that for all w P H it holdsthat 〈w, v〉H “
ş
H〈w, x〉H Nv,Qpdxq.
142 CHAPTER 5. RANDOM VARIABLES
Proof of Theorem 5.3.21. Theorem 5.3.10 proves that there exists an orthonormalbasis B Ď H of H and a globally bounded function λ : B Ñ r0,8q such that for allx P H it holds that
Qx “ÿ
bPB
λb 〈b, x〉H b. (5.128)
Proposition 3.5.5 proves thatř
bPB |λb| ă 8. This and Lemma 3.2.16 show thatthe set λ´1pp0,8qq is at most countable. W.l.o.g. we assume that λ´1pp0,8qq iscountable. Hence, there exist a sequence pbnqnPN Ď λ´1pp0,8q with the propertythat for all n,m P N with n ‰ m it holds that bn ‰ bm and with the property that
tbn P B : n P Nu “ λ´1pp0,8qq. (5.129)
Next let pΩ,F ,Pq be a probability space and let Yn : Ω Ñ R, n P N, be i.i.d. standardnormal random variables. Observe that such a probability space does indeed exist.In the next step let XN : Ω Ñ H, N P N, be the mappings with the property thatfor all N P N it holds that
XN “
Nÿ
n“1
a
λbnYnbn. (5.130)
Note that for all N P N it holds that XN is F/pH, ¨Hq-strongly measurable.Lemma 5.3.19 and the completeness of pH, ¨Hq hence prove that the set
A “ tω P Ω: pXNpωqqNPN Ď H is convergent u (5.131)
is in F . This shows that for all N P N it holds that
Ω Q ω ÞÑ 1Apωq ¨XNpωq P H (5.132)
is F/pH, ¨Hq-strongly measurable. Moreover, observe that for all p P r1,8q it holdsthat
›
›
›
›
supNPN
XN2H
›
›
›
›
LppP;|¨|q
“
›
›
›
›
›
supNPN
«
Nÿ
n“1
λbn |Yn|2
ff›
›
›
›
›
LppP;|¨|q
“
›
›
›
›
›
limNÑ8
«
Nÿ
n“1
λbn |Yn|2
ff›
›
›
›
›
LppP;|¨|q
“ limNÑ8
›
›
›
›
›
Nÿ
n“1
λbn |Yn|2
›
›
›
›
›
LppP;|¨|q
ď limNÑ8
Nÿ
n“1
λbn›
›|Yn|2›
›
LppP;|¨|q“›
›|Y1|2›
›
LppP;|¨|q
8ÿ
n“1
λbn ă 8.
(5.133)
5.3. PROBABILITY MEASURES ON HILBERT SPACES 143
In the next step let X : Ω Ñ H be the mapping with the property that for all ω P Ωit holds that
Xpωq “ v ` 1Apωq”
limNÑ8
XNpωqı
“ v ` limNÑ8
r1Apωq ¨XNpωqs . (5.134)
Combining (5.132) and Theorem 2.4.7 proves that X is F/pH, ¨Hq-strongly mea-surable. Furthermore, note that for all N P N it holds that
E
«
supM1,M2PtN,N`1,... u
XM1 ´XM22H
ff
“ E
«
8ÿ
n“N`1
›
›
›
a
λbnYnbn
›
›
›
2
H
ff
“
8ÿ
n“N`1
E
„
›
›
›
a
λbnYnbn
›
›
›
2
H
“
8ÿ
n“N`1
λbn E“
|Yn|2H
‰
“
8ÿ
n“N`1
λbn ă 8.
(5.135)
This implies that PpAq “ 1. Moreover, (5.133) ensures that for all p P p0,8q it holdsthat
X ´ vHLppP;|¨|q “
›
›
›
›
›
›
›limNÑ8
p1AXNq
›
›
›
H
›
›
›
›
LppP;|¨|q
“
›
›
›limNÑ8
p1A XNHq
›
›
›
LppP;|¨|qď
›
›
›
›
supNPN
XNH
›
›
›
›
LppP;|¨|q
ă 8.
(5.136)
This, in particular, implies that for all p P p0,8q it holds that
E
„
XpH ` supNPN
XNpH
ă 8. (5.137)
Uniform integrability and the fact that PpAq “ 1 hence prove that
E“
X‰
“ E
”
v ` limNÑ8
p1AXNq
ı
“ v `E”
limNÑ8
p1AXNq
ı
“ v ` limNÑ8
Er1AXN s “ v ` limNÑ8
ErXN s
“ v ` limNÑ8
«
Nÿ
n“1
a
λbn ErYns bn
ff
“ v.
(5.138)
In addition, uniform integrability, the fact that PpAq “ 1, and (5.137) ensure that
144 CHAPTER 5. RANDOM VARIABLES
for all x, y P H it holds that
〈x,CovOppXqy〉H “ 〈x,CovOppX ´ vqy〉H “ Er〈x,X ´ v〉H 〈y,X ´ v〉Hs“ lim
NÑ8Er〈x,1AXN〉H 〈y,1AXN〉Hs “ lim
NÑ8Er〈x,XN〉H 〈y,XN〉Hs
“ limNÑ8
〈x,CovOppXNqy〉H “ limNÑ8
«
Nÿ
n“1
λbn 〈x, bn〉H 〈y, bn〉H E“
|Yn|2‰
ff
“ limNÑ8
«
Nÿ
n“1
λbn 〈x, bn〉H 〈y, bn〉H
ff
“ limNÑ8
⟨x,
Nÿ
n“1
λbn 〈bn, y〉H bn
⟩H
“ 〈x,Qy〉H .
(5.139)
This implies thatCovOppXq “ Q. (5.140)
Next note that Lebesgue’s theorem of dominated convergence and the fact thatPpAq “ 1 imply that for all w P H it holds that
`
FHXpPq˘
p〈w, ¨〉Hq “ E“
ei〈w,X〉H‰
“ ei〈w,v〉H E“
ei〈w,X´v〉H‰
“ ei〈w,v〉H limNÑ8
E“
ei〈w,1AXN 〉H‰
“ ei〈w,v〉H limNÑ8
E“
ei〈w,XN 〉H‰
“ ei〈w,v〉H limNÑ8
e´12〈w,CovOppXN qw〉H “ ei〈w,v〉H´
12〈w,CovOppXqw〉H .
(5.141)
This, (5.137), and Corollary 5.3.20 imply that XpPq is a Gaussian measure. Combin-ing this with (5.138) and (5.140) establishes the existence of a probability measureXpPq with the desired properties. The proof of Theorem 5.3.21 is thus completed.
5.3.4.3 Karhunen-Loeve expansion for Brownian motion
Exercise 5.3.22 (An ordinary differential equation of second order). Let a P Rzt0u,T P p0,8q and let v : r0, T s Ñ R be a twice continuously differentiable function withthe property that for all t P r0, T s it holds that v2ptq “ a vptq. Prove that for allt P r0, T s it holds that
˜
vptq
v1ptq
¸
“
˜
1 1?a ´
?a
¸˜
et?a 0
0 e´t?a
¸˜
1 1?a ´
?a
¸´1 ˜
vp0q
v1p0q
¸
(5.142)
and prove that there exist A,B P C such that for all t P r0, T s it holds that vptq “Ae
?at `Be´
?at.
5.3. PROBABILITY MEASURES ON HILBERT SPACES 145
Definition 5.3.23. Let d P N, A P BpRdq. Then we denote by BorelA : BpAq Ñr0,8s the Lebesgue-Borel measure on A.
Let us illustrate Definition 5.3.23 through a simple example. Note that for alla, b, α, β P R with a ď α ď β ď b it holds that
Borelra,bsprα, βsq “ β ´ α. (5.143)
Theorem 5.3.24 (Karhunen-Loeve expansion for Brownian motion). Let pΩ,F ,Pqbe a probability space, let T P p0,8q, let W : r0, T s ˆΩ Ñ R be a standard Brownianmotion with continuous sample paths, let W : Ω Ñ L2pBorelp0,T q; |¨|Rq be the mappingwith the property that for all ω P Ω and Borelp0,T q-almost all t P r0, T s it holds that
pW pωqqptq “ Wtpωq, and let ek P L2pBorelp0,T q; |¨|Rq, k P N, be the vectors withthe property that for all k P N and Borelp0,T q-almost all t P p0, T q it holds that
ekptq “?
2?T
sin`
pk ´ 12qπtT
˘
. Then
(i) W is a Gaussian distributed random variable,
(ii) it holds for all v P L2pBorelp0,T q; |¨|Rq and Borelp0,T q-almost all t P p0, T q that
`
CovOppW q v˘
ptq “
ż T
0
mintt, su vpsq ds “
ż t
0
ż T
s
vpuq du ds
“
ż T
0
ErWtWss vpsq ds,
(5.144)
(iii) it holds that
σP`
CovOppW q˘
“
"
T 2
r12πs2,
T 2
r32πs2,
T 2
r52πs2, . . .
*
“
"
T 2
π2 pk ´ 12q2 P p0,8q : k P N
*
,
(5.145)
(iv) the set tek : k P Nu is an orthonormal basis of L2pBorelp0,T q; |¨|Rq,
(v) it holds for all v P L2pBorelp0,T q; |¨|Rq that
CovOppW q v “8ÿ
n“1
T 2
π2 rn´ 12s2 〈en, v〉L2pBorelp0,T q;|¨|Rq
en, (5.146)
146 CHAPTER 5. RANDOM VARIABLES
(vi) the random variables πTrk ´ 12s xek, W yL2pBorelp0,T q;|¨|Rq
, k P N, are i.i.d. standardnormal random variables, and
(vii) it holds that
W “
8ÿ
k“1
Tπ rk´12s
”
π rk´12s
Txek, W yL2pBorelp0,T q;|¨|Rq
ı
ek. (5.147)
Proof of Theorem 5.3.24. First of all, note that for all v P Cpr0, T s,Rq it holds that
ż T
0
vpsqWs ds “ limNÑ8
«
N´1ÿ
n“0
vpnTNqWnT
N
TN
ff
. (5.148)
This proves that for all v P Cpr0, T s,Rq it holds thatşT
0vpsqWs ds is Gaussian
distributed. This and the fact that Cpr0, T s,Rq is dense in L2pBorelp0,T q; |¨|Rq implies
that for all v P L2pBorelp0,T q; |¨|Rq it holds thatşT
0vpsqWs ds is Gaussian distributed.
It thus holds that W is a Gaussian distributed random variable. Next observe thatfor all v, w P L2pBorelp0,T q; |¨|Rq it holds that
xw,CovOppW qvyL2pBorelp0,T q;|¨|Rq“ E
„ż T
0
vpsqWs ds
ż T
0
wpuqWu du
“
ż T
0
wpuq
ż T
0
ErWsWus vpsq ds du “
ż T
0
wpuq
ż T
0
minps, uq vpsq ds du.
(5.149)
This and Fubini’s theorem show that for all v P L2pBorelp0,T q; |¨|Rq and Borelp0,T q-almost all t P p0, T q it holds that
`
CovOppW q v˘
ptq “
ż T
0
ErWsWts vpsq ds “
ż T
0
mints, tu vpsq ds
“
ż T
0
ż mintu,tu
0
vpuq ds du “
ż t
0
ż T
0
1tsďuu vpuq du ds
“
ż t
0
ż T
s
vpuq du ds.
(5.150)
This proves (ii). In the next step let µ P R and let v : p0, T q Ñ R be an Bpp0, T qq/BpRq-measurable function with the property that
şT
0|vpsq|2 ds “ 1 and with the property
thatµ ¨ v “ CovOppW q v. (5.151)
5.3. PROBABILITY MEASURES ON HILBERT SPACES 147
Equation (5.151) implies that for Borelr0,T s-almost all t P p0, T q it holds that
µ ¨ vptq “
ż T
0
minps, tq vpsq ds. (5.152)
Next let w : r0, T s Ñ R be the function with the property that for all t P r0, T s itholds that
wptq “
ż t
0
ż T
s
vpuq du ds. (5.153)
Note that w is continuously differentiable and observe that w1 : r0, T s Ñ R is abso-lutely continuous. Moreover, note that (ii) implies that for all t P r0, T s it holds that
wptq “
ż T
0
minps, tq vpsq ds. (5.154)
Combining this with (5.152) proves that for Borelr0,T s-almost all t P r0, T s it holdsthat
µ ¨ vptq “ wptq. (5.155)
Next note that (5.155) implies that if µ “ 0, then it holds for all t P r0, T s that0 “ wptq “ w1ptq “ w2ptq and this shows that for Borelr0,T s-almost all t P r0, T s itholds that
0 “ w2ptq “ ´vptq. (5.156)
Equation (5.156) contradicts to the assumption thatşT
0|vpsq|2 ds “ 1 ą 0 and this
proves that µ ‰ 0. Next let v : p0, T q Ñ R be a continuously differentiable functiondefined by vptq :“ 1
µwptq for all t P r0, T s. Equation (5.155) then shows that for
Borelr0,T s-almost all t P p0, T q it holds that
vptq “ vptq (5.157)
and (5.153) and (5.154) hence imply that for all t P r0, T s it holds that
µ ¨ vptq “ wptq “
ż t
0
ż T
s
vpuq du ds “
ż T
0
minps, tq vpsq ds. (5.158)
This proves that w and v are twice continuously differentiable with the property thatfor all t P r0, T s it holds that
v2ptq “1
µw2ptq “
´1
µvptq. (5.159)
148 CHAPTER 5. RANDOM VARIABLES
Exercise 5.3.22 hence proves that there exist A,B P C such that for all t P r0, T s itholds that vptq “ A exp
`
ta
´1µ˘
` B exp`
´ ta
´1µ˘
. This together with the fact
that vp0q “ wp0qµ“ 0 shows that for all t P r0, T s it holds that
vptq “ A”
exp´
ta
´1µ
¯
´ exp´
´ta
´1µ
¯ı
. (5.160)
The identity v1pT q “ w1pT qµ“ 0 and the assumption that
şT
0|vpsq|2 ds “ 1 ą 0 hence
prove that
exp´
2Ta
´1µ
¯
“ ´1. (5.161)
This implies that µ ą 0 and that there exists a k P N such that 2Ta
1µ “ 2πk ´ π.Hence, we obtain that
µ “ T 2
π2rk´12s2 . (5.162)
Putting this into (5.160) proves that for all t P r0, T s it holds that
vptq “ 2iA sin`“
k ´ 12
‰
tπT
˘
. (5.163)
This and the assumption thatşT
0|vpsq|2 ds “ 1 implies that
1 “
ż T
0
|vpsq|2 ds “ 4 |A|2ż T
0
ˇ
ˇsin`“
k ´ 12
‰
sπT
˘ˇ
ˇ
2ds
“ 2T |A|2ż 1
0
ˇ
ˇ
?2 sin
`
rk ´ 12s πs
˘ˇ
ˇ
2ds “ 2T |A|2 .
(5.164)
This and (5.163) prove that there exists a z P t´1, 1u such that for all t P r0, T s itholds that
vptq “ z?
2?T
sin`“
k ´ 12
‰
tπT
˘
“ z ekptq. (5.165)
Furthermore, observe that for all k P N, t P r0, T s it holds that
d2
dt2
„ż T
0
mints, tu ekpsq ds
“d2
dt2
„ż t
0
ż T
s
ekpuq du ds
“d
dt
„ż T
t
ekpsq ds
“ ´ekptq
“ T 2
π2rk´12s2 ¨ e
2kptq “
d2
dt2
”
T 2
π2rk´12s2 ¨ ekptq
ı
.
(5.166)
5.3. PROBABILITY MEASURES ON HILBERT SPACES 149
This together with the fact that for all k P N it holds that
T 2
π2rk´12s2 ¨ ekp0q “ 0 “
ż T
0
minps, 0q ekpsq ds and
T 2
π2rk´12s2 ¨ e
1kpT q “
Tπrk´12s
¨?
2?T¨ cos
`“
k ´ 12
‰
TπT
˘
“ T?
2πrk´12s
?T¨ cos
`“
k ´ 12
‰
π˘
“ 0 “
„
d
dt
ż t
0
ż T
s
ekpuq du ds
t“T
“
„
d
dt
ż T
0
minps, tq ekpsq ds
t“T
(5.167)
proves that for all k P N it holds that
T 2
π2rk´12s2 ¨ ek “ CovOppW q ek. (5.168)
Combining this with (5.162) proves (iii). Moreover, (5.168), (5.165) and Theo-rem 5.3.10 imply (iv) and (v). Finally, (iv), (v) and Corollary 5.3.18 prove (vi)–(vii).The proof of Theorem 5.3.24 is thus completed.
1 function [ Preimage , BM] = KLE Brownian Motion (T,N, Grid )2 Preimage = ( 0 :T/Grid :T) ;3 BM = Preimage ∗0 ;4 for n=1:N5 s q r t e i g e n v n = T/(n ´ 1/2)/ pi ;6 e i g e n f n = sqrt (2/T)∗ sin ( Preimage/ s q r t e i g e n v n ) ;7 BM = BM + s q r t e i g e n v n ∗ e i g e n f n ∗ randn ;8 end9 end
Matlab code 5.1: A Matlab function for approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.
1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 2 ;4 Nodes = 10 ;5 Grid = 2000 ;6 hold on7 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;8 plot ( Preimage ,BM) ;
150 CHAPTER 5. RANDOM VARIABLES
9 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;10 plot ( Preimage ,BM, ’ r ’ ) ;11 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;12 plot ( Preimage ,BM, ’ g ’ ) ;13 hold o f f
Matlab code 5.2: A Matlab code for the approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.
1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 2 ;4 Nodes = 100 ;5 Grid = 2000 ;6 hold on7 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;8 plot ( Preimage ,BM) ;9 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;
10 plot ( Preimage ,BM, ’ r ’ ) ;11 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;12 plot ( Preimage ,BM, ’ g ’ ) ;13 hold o f f
Matlab code 5.3: A Matlab code for approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.
1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 2 ;4 Nodes = 1000 ;5 Grid = 2000 ;6 hold on7 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;8 plot ( Preimage ,BM) ;9 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;
10 plot ( Preimage ,BM, ’ r ’ ) ;11 [ Preimage , BM] = KLE Brownian Motion (T, Nodes , Grid ) ;12 plot ( Preimage ,BM, ’ g ’ ) ;
5.3. PROBABILITY MEASURES ON HILBERT SPACES 151
13 hold o f f
Matlab code 5.4: A Matlab code for approximating the Karhunen-Loewe-Expansion of a one-dimensional Brownian motion.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
−0.5
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
−0.5
0
0.5
1
1.5
2
Figure 5.1: Results of calls of the Matlab codes 5.2–5.4.
152 CHAPTER 5. RANDOM VARIABLES
Chapter 6
Stochastic processes with values ininfinite dimensional spaces
6.1 Hilbert space valued stochastic processes
6.1.1 Standard Wiener processes
Definition 6.1.1. Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, let Q PL1pHq be nonnegative and symmetric, let pΩ,F ,P, pFtqtPr0,T sq be a filtered probabilityspace, let W : r0, T s ˆ Ω Ñ H be an pFtqtPr0,T s-adapted stochastic process with theproperties
(i) that W0 “ 0,
(ii) that W has continuous sample paths,
(iii) that for all t1, t2 P r0, T s with t1 ď t2 it holds that σΩpWt2 ´Wt1q and Ft1 areindependent, and
(iv) that for all t1, t2 P r0, T s with t1 ď t2 it holds that pWt2 ´Wt1qpPq “ N0,Qpt2´t1q.
Then W is called a standard Q-Wiener process with respect to (w.r.t.) pFtqtPr0,T s.
Definition 6.1.2. Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, letQ P L1pHq be nonnegative and symmetric, let pΩ,F ,Pq be a probability space, andlet W : r0, T s ˆ Ω Ñ H be a standard Q-Wiener process w.r.t. pFWt qtPr0,T s. Then Wis called a standard Q-Wiener process.
153
154 CHAPTER 6. STOCHASTIC PROCESSES
Theorem 6.1.3. Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, andlet Q P L1pHq be nonnegative and symmetric. Then there exist a probability spacepΩ,F ,Pq and a standard Q-Wiener process W : r0, T s ˆ Ω Ñ H.
Theorem 6.1.3 can, e.g., be proved by using a Karhunen-Loeve expansion similaras in the proof of Theorem 5.3.21.
6.1.2 Pseudo inverse
Lemma 6.1.4. Let K P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbert
spaces, and let A P LpH1, H2q. Then the mapping A|KernpAqJ : KernpAqJ Ñ H2 isinjective and it holds that impAq “ im
`
A|KernpAqJ˘
.
Proof of Lemma 6.1.4. First of all, recall that
KernpAqJ “
v P H1 :“
@u P KernpAq : 〈v, u〉H1“ 0
‰(
(6.1)
is a K-vector subspace of H1. The mapping A|KernpAqJ : KernpAqJ Ñ H2 is thus alinear mapping from KernpAqJ to H2. It thus holds that
A|KernpAqJ is injective ô Kern`
A|KernpAqJ˘
“ t0u. (6.2)
Next note that
Kern`
A|KernpAqJ˘
“
v P KernpAqJ : A|KernpAqJpvq “ 0(
“
v P KernpAqJ : Av “ 0(
“
v P KernpAqJ : v P KernpAq(
“ KernpAqJ XKernpAq “ t0u.
(6.3)
Combining this with (6.2) proves that A|KernpAqJ is injective. Moreover, observe that
impAq “ ApH1q “ tAv P H2 : v P H1u
“
A“
PKernpAq rvs ` PKernpAqJ rvs‰
P H2 : v P H1
(
“
APKernpAqJ rvs P H2 : v P H1
(
“
APKernpAqJ rvs P H2 : v P KernpAqJ(
“
Av P H2 : v P KernpAqJ(
“ im`
A|KernpAqJ˘
.
(6.4)
The proof of Lemma 6.1.4 is thus completed.
6.1. HILBERT SPACE VALUED STOCHASTIC PROCESSES 155
Lemma 6.1.4 allows us to introduce the following concept.
Definition 6.1.5. Let K P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbertspaces, and let A P LpH1, H2q. Then we denote by A´1 : impAq Ñ H1 the linearoperator with the property that for all v P impAq it holds that
A´1pvq “ A|´1
KernpAqJpvq (6.5)
and we call A´1 the pseudo inverse of A.
The next exercise will help us to get more familiar with the pseudo inverse.
Exercise 6.1.6. Let A : L2pBorelp0,1q; |¨|Rq Ñ R be the linear mapping with the prop-erty that for all v P L2pBorelp0,1q; |¨|Rq it holds that
Av “
ż 1
0
vpxq dx. (6.6)
Specify DpA´1q, impA´1q, rangepA´1q, and A´1v, v P DpA´1q, explicity. Show thatyour specifications are indeed correct.
In the following proposition we present an important property of the pseudoinverse of a bounded linear operator.
Proposition 6.1.7 (Minimality property of the pseudo inverse). Let K P tR,Cu,let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbert spaces, let A P LpH1, H2q, and letv P impAq “ DpA´1q. Then A´1vH1
“ infuPA´1ptvuq uH1and
A´1v(
“
#
w P H1 :
«
Aw “ v and wH1“ inf
uPH1,Au“v
uH1
ff+
“
"
w P A´1ptvuq : wH1
“ infuPA´1ptvuq
uH1
*
“ A´1ptvuq X
“
KernpAqJ‰
.
(6.7)
Proof of Proposition 6.1.7. First of all, note that Lemma 6.1.4 and the definition ofthe pseudo inverse prove that
A´1v(
“
!
A|´1KernpAqJ
pvq)
“
w P KernpAqJ : Aw “ v(
““
KernpAqJ‰
X A´1ptvuq.
(6.8)
156 CHAPTER 6. STOCHASTIC PROCESSES
Next recall that KernpAq Ď H1 is a closed subspace of H1. Definition 3.3.12 thusshows that for all w P A´1ptvuq “ tu P H1 : Au “ vu it holds that
A|KernpAqJ“
PKernpAqJ rws‰
“ A“
PKernpAqJ rws‰
“ A“
w ´ PKernpAq rws‰
“ Aw “ v.(6.9)
The fact that A|KernpAqJ is injective (see Lemma 6.1.4) hence proves that for allw P A´1ptvuq “ tu P H1 : Au “ vu it holds that
PKernpAqJ rws “ A´1v. (6.10)
This implies that for all w P A´1ptvuq “ tu P H1 : Au “ vu it holds that
wH1“›
›PKernpAq rws ` PKernpAqJ rws›
›
H1
“
b
›
›PKernpAq rws›
›
2
H1`›
›PKernpAqJ rws›
›
2
H1
“
b
›
›PKernpAq rws›
›
2
H1` A´1pvq2H1
ě›
›A´1pvq
›
›
2
H1.
(6.11)
This and the fact that A´1pvq P A´1ptvuq prove that
infuPA´1ptvuq
uH1“›
›A´1pvq
›
›
H1. (6.12)
This, (6.11), and (6.8) imply that"
w P A´1ptvuq : wH1
“ infuPA´1ptvuq
uH1
*
“
!
w P A´1ptvuq : wH1
“›
›A´1pvq
›
›
H1
)
“
"
w P A´1ptvuq :
b
›
›PKernpAq rws›
›
2
H1` A´1pvq2H1
“›
›A´1pvq
›
›
H1
*
“
!
w P A´1ptvuq :
›
›PKernpAq rws›
›
2
H1“ 0
)
“
w P A´1ptvuq : PKernpAq rws “ 0
(
“ A´1ptvuq X
“
KernpAqJ‰
“
A´1pvq
(
.
(6.13)
This, (6.8), and (6.12) complete the proof of Proposition 6.1.7.
The pseudo inverse allows us to define a Hilbert space structure on the imageof a bounded linear operator on Hilbert spaces. This is the subject of the nextproposition.
Proposition 6.1.8 (Image Hilbert space). Let K P tR,Cu, let pHk, 〈¨, ¨〉Hk , ¨Hkq,k P t1, 2u, be K-Hilbert spaces, and let A P LpH1, H2q. Then it holds that the triple`
impAq, 〈A´1p¨q, A´1p¨q〉H1, A´1p¨qH1
˘
is an K-Hilbert space.
6.2. STOCHASTIC INTEGRATION 157
Proof of Proposition 6.1.8. First of all, note that the fact that A´1 : impAq Ñ H1 isa linear operator implies that the triple
`
impAq, 〈A´1p¨q, A´1p¨q〉H1, A´1p¨qH1
˘
is aK-inner product space. It thus remains to prove that pimpAq, A´1p¨qH1
q is com-plete. To see this let pvnqnPN Ď impAq be a Cauchy sequence in pimpAq, A´1p¨qH1
q.Then A´1pvnq P KernpAqJ, n P N, is a Cauchy sequence in pKernpAqJ, ¨H1
q. Com-pleteness of pKernpAqJ, ¨H1
q hence proves that there exists a vector w P KernpAqJ
such that limnÑ8 w ´ A´1pvnqH1
“ 0. This proves that Aw P impAq satisfieslimnÑ8 A
´1pAw ´ vnqH1“ 0. The proof of Proposition 6.1.8 is thus completed.
6.2 Stochastic integration with respect to infinite
dimensional Wiener processes
The following presentations are similiar to the presentations in Section 2.3 in Prevot& Rockner [24] and in Chapter 3 in the lecture notes of the course Numerical Analysisof Stochastic Ordinary Differential Equations.
6.2.1 Filtrations
Definition 6.2.1 (Filtration). Let pΩ,Fq be a measurable space, let T Ď`
R Y
t´8,8u˘
be a set, and let pFtqtPT be a family of sigma-algebras on Ω with the propertythat for every t1, t2 P T with t1 ď t2 it holds that
Ft1 Ď Ft2 Ď F . (6.14)
Then pFtqtPT is called a filtration on pΩ,Fq.
Definition 6.2.2 (Filtrations associated to a filtration). Let T Ď`
RYt´8,8u˘
bea set and let pΩ,Fq be a measurable space with a filtration pFtqtPT. Then we denoteby pF´t qtPT and pF`t qtPT the filtrations on pΩ,Fq with the property that for all t P Tit holds that
F´t “
#
σΩ
`
YsPTXp´8,tqFs˘
: t ą infpTqFt : t “ infpTq
(6.15)
and
F`t “
#
XsPTXpt,8qFs : t ă suppTqFt : t “ suppTq
. (6.16)
158 CHAPTER 6. STOCHASTIC PROCESSES
Observe, in the setting of Definition 6.2.2, that for all t P T it holds that F´t ĎFt Ď F`t .
Definition 6.2.3. Let T Ď`
R Y t´8,8u˘
be a set and let pΩ,Fq be a measurablespace with a filtration pFtqtPT which satisfies that for all t P T it holds that Ft “F´t (Ft “ F`t ). Then we say that the filtration pFtqtPT is left-continuous (right-continuous).
Next we present the notion of a normal filtration (cf., e.g., Definition 2.1.11 in[24]) and of a stochastic basis (cf. Appendix E in [24]).
Definition 6.2.4 (Normal filtration). Let T P r0,8q and let pΩ,F ,Pq be a probabilityspace with a filtration pFtqtPr0,T s which satisfies
(i) that tA P F : P pAq “ 0u Ď F0 and
(ii) that pFtqtPr0,T s is right-continuous.
Then we say that pFtqtPr0,T s is normal (or also that pFtqtPr0,T s fulfills the usual con-ditions).
Definition 6.2.5 (Stochastic basis). Let T P r0,8q and let pΩ,F , P q be a probabilityspace with a normal filtration pFtqtPr0,T s. Then the quadrupel pΩ,F ,P, pFtqtPr0,T sq iscalled a stochastic basis.
Let us also point out that if T P r0,8q is a real number and if pΩ,F , P qis a probability space with a filtration pFtqtPr0,T s, then sometimes the quadrupelpΩ,F ,P, pFtqtPr0,T sq is called a stochastic basis in the literature although pFtqtPr0,T s isnot necessarily normal.
6.2.2 Lenglart’s inequality
Definition 6.2.6 (Random time). Let T Ď pRY t´8,8uq be a set, let pΩ,F ,Pq bea probability space, and let τ : Ω Ñ T be an F/BpTq-measurable mapping. Then τ iscalled a random time.
Observe, in the setting of Definition 6.2.6, that for every t P T it holds thattτ ď tu P F .
Definition 6.2.7 (Stopping time). Let T Ď pR Y t´8,8uq be a set, let pΩ,F ,Pqbe a probability space with a filtration pFtqtPT, and let τ : Ω Ñ T be a mapping withthe property that for all t P T it holds that tτ ď tu P Ft. Then τ is called an pFtqtPT-stopping time.
6.2. STOCHASTIC INTEGRATION 159
A stopping time on a filtered probability space also induces a sigma-algebra. Thisis the subject of the next definition.
Definition 6.2.8. Let T Ď pR Y t´8,8uq be a set, let pΩ,F ,Pq be a probabilityspace with a filtration pFtqtPT, and let τ : Ω Ñ T be an pFtqtPT-stopping time. Thenwe denote by Fτ the set given by
Fτ “ tA P pYtPTFtq : p@ t P T : AX tτ ď tu P Ftqu (6.17)
and we call Fτ the sigma-algebra at the stopping time τ .
Exercise 6.2.9. Let T Ď pRYt´8,8uq be a set, let pΩ,F ,Pq be a probability spacewith a filtration pFtqtPT, and let τ, ρ : Ω Ñ T be pFtqtPT-stopping times. Prove thenthat mintτ, ρu is an pFtqtPT-stopping time.
In (6.20) in the following result, Proposition 6.2.10, we prove a powerful inequalitywhich is known as Lenglart inequality in the literature. Proposition 6.2.10 and itsproof are extensions of Problem 1.4.15, Remark 1.4.17 and Solution 4.15 in Section1.6 in [16].
Proposition 6.2.10 (Lenglart inequality). Let pΩ,F ,Pq be a probability space witha filtration pFtqtPr0,8q, let X, Y : r0,8qˆΩ Ñ r0,8q be pFtqtPr0,8q-adapted stochasticprocesses with continuous sample paths such that for all bounded pFtqtPr0,8q-stoppingtimes τ : Ω Ñ r0,8q it holds that E
“
Xτ
‰
ď E“
suptPr0,τ s Yt‰
. Then for all ε, δ P p0,8qand all pFtqtPr0,8q-stopping times τ : Ω Ñ r0,8q it holds that
P`
suptPr0,τ sXt ě ε˘
ď 1εE“
suptPr0,τ s Yt‰
, (6.18)
P`
suptPr0,τ sXt ě ε, suptPr0,τ s Yt ă δ˘
ď 1εE“
min
δ, suptPr0,τ s Yt(‰
, (6.19)
P`
suptPr0,τ sXt ě ε˘
ď 1εE“
min
δ, suptPr0,τ s Yt(‰
` P`
suptPr0,τ s Yt ě δ˘
, (6.20)
E“
min
ε, suptPr0,τ sXt
(‰
ď
”
2?ε` ε?
δ
ı
ˇ
ˇE“
min
δ, suptPr0,τ s Yt(‰ˇ
ˇ
12, (6.21)
E“
min
1, suptPr0,τ sXt
(‰
ď 3ˇ
ˇE“
min
1, suptPr0,τ s Yt(‰ˇ
ˇ
12. (6.22)
Proof of Proposition 6.2.10. Throughout this proof let ρXε : Ω Ñ r0,8s, ε P r0,8q,and ρYε : Ω Ñ r0,8s, ε P r0,8q, be the mappings with the property that for allε P r0,8q it holds that
ρXε “ inf`
t P r0,8q : Xt ě ε(
Y t8u˘
, (6.23)
160 CHAPTER 6. STOCHASTIC PROCESSES
ρYε “ inf`
t P r0,8q : supsPr0,ts Ys ě ε(
Y t8u˘
. (6.24)
Then observe that for all ε P r0,8q, n P N and all pFtqtPr0,8q-stopping times τ : Ω Ñr0,8q it holds that
εP`
suptPr0,mintτ,nusXt ě ε˘
“ εP´
D t P r0,mintτ, nus : Xt ě ε¯
“ εP´!
D t P r0,mintτ, nus : Xt ě ε)
X
!
ρXε ď mintτ, nu)¯
“ εP´!
D t P r0,mintτ, nus : Xt ě ε)
X
!
ρXε ď mintτ, nu)
X
!
Xmintτ,n,ρXε uě ε
)¯
ď εP´
Xmintτ,n,ρXε uě ε
¯
“ E
”
ε1tXmintτ,n,ρXε u
ěεu
ı
ď E
”
Xmintτ,n,ρXε u1tX
mintτ,n,ρXε uěεu
ı
ď E“
Xmintτ,n,ρXε u
‰
.
(6.25)
Combining this with the fact for all ε P r0,8q, n P N and all pFtqtPr0,8q-stoppingtimes τ : Ω Ñ r0,8q it holds that mintτ, n, ρXε u is a bounded pFtqtPr0,8q-stopping time(see Exercise 6.2.9) ensures that for all ε P r0,8q, n P N and all pFtqtPr0,8q-stoppingtimes τ : Ω Ñ r0,8q it holds that
εP`
suptPr0,mintτ,nusXt ě ε˘
ď E“
Xmintτ,n,ρXε u
‰
ď E“
suptPr0,mintτ,n,ρXε usYt‰
ď E“
suptPr0,τ s Yt‰
.(6.26)
Hence, we obtain that for all ε P r0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ
r0,8q it holds that
εP`
suptPr0,τ sXt ě ε˘
“ εP`
YnPN
suptPr0,mintτ,nusXt ě ε(˘
“ ε limnÑ8
P`
suptPr0,mintτ,nusXt ě ε˘
ď E“
suptPr0,τ s Yt‰
.(6.27)
This proves (6.18). In the next step we observe that (6.18) ensures that for allε, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ p0,8q it holds that
P`
suptPr0,τ sXt ě ε, suptPr0,τ s Yt ă δ˘
“ P`
suptPr0,τ sXt ě ε, ρYδ ą τ, suptPr0,τ s Yt ă δ˘
“ P´
suptPr0,mintτ,ρYδ usXt ě ε, ρYδ ą τ, suptPr0,τ s Yt ă δ
¯
ď P´
suptPr0,mintτ,ρYδ usXt ě ε
¯
ď 1εE
”
suptPr0,mintτ,ρYδ usYt
ı
“ 1εE
”
min
δ, suptPr0,mintτ,ρYδ usYt(
ı
ď 1εE“
min
δ, suptPr0,τ s Yt(‰
.
(6.28)
6.2. STOCHASTIC INTEGRATION 161
This proves (6.19). Furthermore, we observe that (6.19) shows that for all ε, δ Pp0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ p0,8q it holds that
P`
suptPr0,τ sXt ě ε˘
ď P`
suptPr0,τ sXt ě ε, suptPr0,τ s Yt ă δ˘
` P`
suptPr0,τ s Yt ě δ˘
ď 1εE“
min
δ, suptPr0,τ s Yt(‰
` P`
suptPr0,τ s Yt ě δ˘
.
(6.29)
This proves (6.20). Next we note that (6.20) and the Markov inequality show thatfor all r, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ p0,8q it holds that
P`
suptPr0,τ sXt ě r˘
ď 1rE“
min
δ, suptPr0,τ s Yt(‰
` P`
min
δ, suptPr0,τ s Yt(
ě δ˘
ď“
1r` 1
δ
‰
E“
min
δ, suptPr0,τ s Yt(‰
.
(6.30)
This implies that for all ε, δ, r P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñ
p0,8q it holds that
E“
min
ε, suptPr0,τ sXt
(‰
“ E
”
min
ε, suptPr0,τ sXt
(
1tsuptPr0,τsXtăru
ı
`E
”
min
ε, suptPr0,τ sXt
(
1tsuptPr0,τsXtěru
ı
ď mintε, ru ` εP`
suptPr0,τ sXt ě r˘
ď mintε, ru ` ε“
1r` 1
δ
‰
E“
min
δ, suptPr0,τ s Yt(‰
ď r ` ε“
1r` 1
δ
‰
E“
min
δ, suptPr0,τ s Yt(‰
.
(6.31)
Hence, we obtain that for all ε, δ P p0,8q and all pFtqtPr0,8q-stopping times τ : Ω Ñp0,8q it holds that
E“
min
ε, suptPr0,τ sXt
(‰
ď infrPp0,8q
`
r ` εrE“
min
δ, suptPr0,τ s Yt(‰
` εδE“
min
δ, suptPr0,τ s Yt(‰˘
ďˇ
ˇεE“
min
δ, suptPr0,τ s Yt(‰ˇ
ˇ
12
`?εˇ
ˇE“
min
δ, suptPr0,τ s Yt(‰ˇ
ˇ
12` ε
δE“
min
δ, suptPr0,τ s Yt(‰
.
(6.32)
This proves (6.21). Moreover, we note that (6.22) is an immediate consequence of(6.21). The proof of Proposition 6.2.10 is thus completed.
162 CHAPTER 6. STOCHASTIC PROCESSES
Exercise 6.2.11 (Characterization of convergence in probability). Let pΩ,F ,Pq bea probability space, let pE, dEq be a metric space, and let Xn : Ω Ñ E, n P N0, beF/pE, dEq-strongly measurable mappings. Then the following three statements areequivalent:
(i) For all c P p0,8q it holds that
limnÑ8
E“
mintc, dEpX0, Xnqu‰
“ 0. (6.33)
(ii) There exists a c P p0,8q such that
limnÑ8
E“
mintc, dEpX0, Xnqu‰
“ 0. (6.34)
(iii) For all ε P p0,8q it holds that
limnÑ8
PpdEpX0, Xnq ą εq “ 0. (6.35)
6.2.3 Modifications and indistinguishability
This material is an extended version from Barth et al. 2014. Next we address twonotions that somehow describe when two stochastic processes are “equal up to setsof measure zero”.
Definition 6.2.12 (Modifications). Let pΩ,F ,Pq be a probability space, let pS,Sqbe a measurable space, let T Ď R be a set, and let X, Y : T ˆ Ω Ñ S be stochasticprocesses such that for every t P T it holds that there exists an event A P F withPpAq “ 1 and
A Ď tXt “ Ytu. (6.36)
Then X and Y are called modifications of each other (i.e., X is called a modificationof Y and Y is called a modification of X).
Exercise 6.2.13. Prove or disprove the following statement: For all measurablespaces pΩ,Fq it holds that tpω, ωq P Ω2 : ω P Ωu P F b F .
Exercise 6.2.14. Specify explicitly measurable spaces pΩ,Fq and pS,Sq and F/S-measurable mappings X, Y : Ω Ñ S such that tX “ Y u “ tω P Ω: Xpωq “ Y pωqu RF . Prove that your result is correct.
6.2. STOCHASTIC INTEGRATION 163
Definition 6.2.15 (Indistinguishablility). Let pΩ,F ,Pq be a probability space, letpS,Sq be a measurable space, let T Ď R be a set, and let X, Y : T ˆ Ω Ñ S bestochastic processes with the property that there exists an event A P F with PpAq “ 1and
A Ď pXtPTtXt “ Ytuq . (6.37)
Then X and Y are called indistinguishable from each other (i.e., X is called indis-tinguishable from Y and Y is called indistinguishable from X).
Let us illustrate Definitions 6.2.12 and 6.2.15 through a simple example (see, e.g.,Kuhn [20]).
Example 6.2.16. Let pΩ,F ,Pq be a probability space, let pFtqtPr0,1s be the filtrationon pΩ,Fq with the property that for all t P r0, 1s it holds that Ft “ F , let τ : Ω Ñ r0, 1sbe an F/Bpr0, 1sq-measurable mapping such that τpPq “ Borelr0,1s, let X, Y : r0, 1s ˆΩ Ñ R be the functions with the property that for all ω P Ω, t P r0, 1s it holds that
Xtpωq “ 0 and Ytpωq “
#
1 : t “ τpωq
0 : t ‰ τpωq. (6.38)
Then
(i) it holds that X, Y are pFtqtPr0,T s-predictable stochastic processes (indeed, letY n : r0, T s ˆ Ω Ñ R, n P N, be the mappings with the property that for alln P N, t P r0, T s it holds that Y n
t pωq “ 1pτpωq´1n,τpωqsptq, observe that forall n P N it holds that Y n is pFtqtPr0,T s-predictable and note that @ pt, ωq Pr0, T s ˆ Ω: limnÑ8 Y
nt pωq “ Ytpωq),
(ii) it holds that τ is an pFtqtPr0,T s-stopping time,
(iii) it holds for all ω P Ω that Xτpωqpωq “ 0 ‰ 1 “ Yτpωqpωq,
(iv) it holds that!
ω P Ω:`
@ t P r0, T s : Xtpωq “ Ytpωq˘
)
“
!
ω P Ω:`
@ t P r0, T s : Ytpωq “ 0˘
)
“ H,(6.39)
(v) it holds for all t P r0, T s that
PpXt “ Ytq “ PpYt “ 0q “ Ppτ ‰ tq “ 1, (6.40)
(vi) and it holds that X and Y are modification of each other but X and Y are notindistinguishable from of each other.
164 CHAPTER 6. STOCHASTIC PROCESSES
6.2.4 Predictability
Definition 6.2.17 (Predictable sigma-algebra). Let T P r0,8q and let pΩ,Fq be ameasurable space with a filtration pFtqtPr0,T s. Then we denote by Pred
`
pFtqtPr0,T s˘
thesigma-algebra given by
Pred`
pFtqtPr0,T s˘
“
σr0,T sˆΩ
´
tps, ts ˆ A : A P Fs and s, t P r0, T s with s ă tu Y tt0u ˆ A : A P F0u
¯
(6.41)
and we call Pred`
pFtqtPr0,T s˘
the predictable sigma-algebra of pFtqtPr0,T s.
Note, in the setting of Definition 6.2.17, that the definition of the sigma-algebraPred
`
pFtqtPr0,T s˘
depends on the filtration pFtqtPr0,T s.
Definition 6.2.18 (Predictability). Let T P r0,8q, let pS,Sq be a measurable space,let pΩ,Fq be a measurable space with a filtration pFtqtPr0,T s, and let X : r0, T s ˆΩ ÑS be an Pred
`
pFtqtPr0,T s˘
/S-measurable mapping. Then X is called an pFtqtPr0,T s-predictable (stochastic) process (Then X is called pFtqtPr0,T s-predictable).
Observe that if T P p0,8q and if pΩ,Fq is a measurable space with a filtrationpFtqtPr0,T s, then
PredppFtqtPr0,T sq Ď σr0,T sˆΩ
`
B ˆ A : B P Bpr0, T sq and A P FT˘
“ Bpr0, T sq b FT .(6.42)
This is fact is used in the next definition.
Definition 6.2.19 (Product measure on the predictable sigma-algebra). Let T P
p0,8q and let pΩ,F ,Pq be a probability space with a filtration pFtqtPr0,T s. Then wedenote by
PP,pFtqtPr0,T s : PredpP, pFtqtPr0,T sq Ñ r0,8q (6.43)
the measure given by
PP,pFtqtPr0,T s “ pBorelr0,T sbPq|PredppFtqtPr0,T sq. (6.44)
Let T P p0,8q and let pΩ,F ,Pq be a probability space with a filtration pFtqtPr0,T s.Then we note that for all t1, t2 P r0, T s, A P Ft1 with t1 ă t2 it holds that
PP,pFtqtPr0,T sppt1, t2s ˆ Aq “ pt2 ´ t1q ¨ PpAq. (6.45)
6.2. STOCHASTIC INTEGRATION 165
6.2.5 Construction of the stochastic integral
In the next step we introduce the notion of an elementary process. For this we recallthe notion of a simple function; see Definition 2.3.1 above.
Definition 6.2.20 (Elementary process). Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq andpU, 〈¨, ¨〉U , ¨Uq be R-Hilbert spaces, let pΩ,Fq be a measurable space with a filtrationpFtqtPr0,T s, and let X : r0, T s ˆ Ω Ñ LpU,Hq be a mapping with the property thatthere exist n P N, 0 ď t1 ă . . . ă tn ď T and for every k P t1, . . . , n ´ 1u anFtk/BpLpU,Hqq-simple function Yk : Ω Ñ LpU,Hq such that for all t P r0, T s it holdsthat
Xt “
n´1ÿ
k“1
Yk ¨ 1ptk,tk`1sptq . (6.46)
Then X is called pFtqtPr0,T s-elementary (or just elementary).
Elementary processes in the sense of Definition 6.2.20 are predictable in the senseof Definition 6.2.18. Let pU, 〈¨, ¨〉U , ¨Uq be an R-Hilbert space, let T P r0,8q, letpΩ,F ,Pq be a probability space, let Q P L1pUq be nonnegative and symmetric, andlet W : r0, T sˆΩ Ñ U be a standard Q-Wiener process. In the stochastic integrationtheory with respect to the possibly infinite dimensional standard Q-Wiener processW the Hilbert space
´
Q12pUq,
⟨Q´
12p¨q, Q´
12p¨q⟩U,›
›Q´12p¨q›
›
2
U
¯
(6.47)
plays an important role. Recall that Q12 is defined according to Theorem 3.4.12and Q´12 is the pseudo inverse of Q12 (see Definition 6.1.5 above). According toProposition 6.1.8 the triple (6.47) is indeed an R-Hilbert space. Lemma 6.2.23 belowillustrate the appearence of the Hilbert space in (6.47) in the stochastic integrationtheory. Before we present Lemma 6.2.23, we note the following exercise and itspreceding remark.
Exercise 6.2.21 (Embedding of LpU,Hq intoHSpQ12pUq, Hq). Let pH, 〈¨, ¨〉H , ¨Hqand pU, 〈¨, ¨〉U , ¨Uq be R-Hilbert spaces, let Q P L1pUq be nonnegative and symmet-ric, and let A P LpU,Hq. Prove then that A|Q12pUq P HSpQ12pUq, Hq and that
›
›A|Q12pUq
›
›
HSpQ12pUq,Hq“›
›AQ12›
›
HSpU,Hqď ALpU,Hq
›
›Q12›
›
HSpUqă 8. (6.48)
166 CHAPTER 6. STOCHASTIC PROCESSES
Remark 6.2.22 (Embedding of LpU,Hq into HSpQ12pUq, Hq). Let pH, 〈¨, ¨〉H , ¨Hqand pU, 〈¨, ¨〉U , ¨Uq be R-Hilbert spaces, let Q P L1pUq be nonnegative and symmet-ric, and let A P LpU,Hq. Then we often simply write A as an abbreviation forA|Q12pUq.
We are now ready to present Lemma 6.2.23, which illustrates the appearence ofthe Hilbert space in (6.47) in the stochastic integration theory.
Lemma 6.2.23. Let T P r0,8q, s, t P r0, T s with s ď t, let pΩ,F ,P, pFtqtPr0,T q bea stochastic basis, let pU, 〈¨, ¨〉U , ¨Uq be a separable R-Hilbert space, let A : Ω Ñ
LpU,Hq be an Fs/BpLpU,Hqq-simple function, let Q P L1pUq be nonnegative andsymmetric, and let W : r0, T s ˆ Ω Ñ U be a standard Q-Wiener process w.r.t.pFtqtPr0,T s. Then
E“
A pWt ´Wsq2H
‰
“ E
”
A2HSpQ12pUq,Hq
ı
pt´ sq . (6.49)
Proof of Lemma 6.2.23. Let B Ď U be an orthonormal basis of U and let λ : B Ñr0,8q be a globally bounded function such that for all u P U it holds that
Qu “ÿ
bPB
λb 〈b, u〉U b. (6.50)
Then note that the fact that B Ď U is an orthonormal basis of U and the continuityof A imply that
E“
A pWt ´Wsq2H
‰
“ E
»
–
›
›
›
›
›
A
˜
ÿ
bPB
〈b,Wt ´Ws〉U b
¸›
›
›
›
›
2
H
fi
fl
“ E
»
–
›
›
›
›
›
ÿ
bPB
〈b,Wt ´Ws〉U Ab
›
›
›
›
›
2
H
fi
fl
“ E
«
ÿ
b1,b2PB
〈b1,Wt ´Ws〉U 〈b2,Wt ´Ws〉U 〈Ab1, Ab2〉H
ff
“ÿ
b1,b2PB
Er〈b1,Wt ´Ws〉U 〈b2,Wt ´Ws〉U 〈Ab1, Ab2〉Hs .
(6.51)
6.2. STOCHASTIC INTEGRATION 167
Independency and the definition of a standard Q-Wiener process hence implies that
E“
A pWt ´Wsq2H
‰
“ÿ
b1,b2PB
Er〈b1,Wt ´Ws〉U 〈b2,Wt ´Ws〉U sEr〈Ab1, Ab2〉Hs
“ÿ
bPB
E“
|〈b,Wt ´Ws〉U |2‰
E“
Ab2H‰
“ pt´ sqÿ
bPB
〈b,Qb〉U E“
Ab2H‰
“ pt´ sqÿ
bPλ´1pp0,8qq
〈b,Qb〉Ulooomooon
“λb
E“
Ab2H‰
.
(6.52)
This proves that
E“
A pWt ´Wsq2H
‰
“ pt´ sqE
»
–
ÿ
bPλ´1pp0,8qq
›
›
›Aa
λb b›
›
›
2
H
fi
fl
“ pt´ sqE
»
–
ÿ
bPλ´1pp0,8qq
›
›AQ12b›
›
2
H
fi
fl .
(6.53)
Next we observe that the set
Q12pbq P im
`
Q12˘
: b P λ´1pp0,8qq
(
(6.54)
is an orthonormal basis in the R-Hilbert space´
Q12pUq,
⟨Q´
12p¨q, Q´
12p¨q⟩U,›
›Q´12p¨q›
›
2
U
¯
. (6.55)
Combining this with (6.53) completes the proof of Lemma 6.2.23.
Lemma 6.2.24 (Ito’s isometry in infinite dimension for elementary processes). Letn P t2, 3, . . . u, T P r0,8q, 0 ď t1 ă ¨ ¨ ¨ ă tn “ T , let pΩ,F ,P, pFtqtPr0,T q bea stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be separable R-Hilbertspaces, for every k P t1, 2, . . . , n ´ 1u let Yk : Ω Ñ LpU,Hq be an Ftk/BpLpU,Hqq-simple function, let Q P L1pUq be nonnegative and symmetric, and let W : r0, T s ˆΩ Ñ U be a standard Q-Wiener process w.r.t. pFtqtPr0,T s. Then
E
»
–
›
›
›
›
›
n´1ÿ
k“1
Yk`
Wtk`1´Wtk
˘
›
›
›
›
›
2
H
fi
fl “ E
«
n´1ÿ
k“1
Yk2HSpQ12pUq,Hq ptk`1 ´ tkq
ff
“
ż T
0
E
»
–
›
›
›
›
›
n´1ÿ
k“1
Yk 1ptk,tk`1spsq
›
›
›
›
›
2
HSpQ12pUq,Hq
fi
fl ds.
(6.56)
168 CHAPTER 6. STOCHASTIC PROCESSES
Proof of Lemma 6.2.24. Note that
E
»
–
›
›
›
›
›
n´1ÿ
k“1
Yk`
Wtk`1´Wtk
˘
›
›
›
›
›
2
H
fi
fl
“
n´1ÿ
k,l“1
E“⟨Yk
`
Wtk`1´Wtk
˘
, Yl`
Wtl`1´Wtl
˘⟩H
‰
“ 2ÿ
k,lPt1,...,n´1ukăl
E“⟨rYls
˚Yk`
Wtk`1´Wtk
˘
,Wtl`1´Wtl
⟩U
‰
`
n´1ÿ
k“1
E
”
›
›Yk`
Wtk`1´Wtk
˘›
›
2
H
ı
.
(6.57)
Independency and Lemma 6.2.23 hence imply that for all orthonormal bases B Ď Uof U it holds that
E
»
–
›
›
›
›
›
n´1ÿ
k“1
Yk`
Wtk`1´Wtk
˘
›
›
›
›
›
2
H
fi
fl
“ 2ÿ
k,lPt1,...,n´1ukăl
E
«
ÿ
bPB
⟨b, rYls
˚Yk`
Wtk`1´Wtk
˘⟩U
⟨b,Wtl`1
´Wtl
⟩U
ff
`
n´1ÿ
k“1
E
”
Yk2HSpQ12pUq,Hq
ı
ptk`1 ´ tkq
“ 2ÿ
k,lPt1,...,n´1ukăl
ÿ
bPB
E“⟨b, rYls
˚Yk`
Wtk`1´Wtk
˘⟩U
‰
E“⟨b,Wtl`1
´Wtl
⟩U
‰
`
n´1ÿ
k“1
E
”
Yk2HSpQ12pUq,Hq
ı
ptk`1 ´ tkq “n´1ÿ
k“1
E
”
Yk2HSpQ12pUq,Hq
ı
ptk`1 ´ tkq .
(6.58)
The proof of Lemma 6.2.24 is thus completed.
The random variableřn´1k“1 Yk
`
Wtk`1´Wtk
˘
in (6.56) will be the stochastic in-
tegral of the elementary stochastic processřn´1k“1 Yk 1ptk,tk`1s in (6.56). Our aim is to
integrate more general stochastic processes. To do so the following lemma is crucial.
6.2. STOCHASTIC INTEGRATION 169
Lemma 6.2.25 (Density). Let T P r0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be R-Hilbert spaces, let Q P L1pUqbe nonnegative and symmetric, and let X P L0
`
PP,pFtqtPr0,T s ; ¨HSpQ12pUq,Hq
˘
satisfy
that it holds P-a.s. thatşT
0Xs
2HSpQ12pUq,Hq
ds ă 8. Then there exists a sequence
Xn : r0, T s ˆΩ Ñ LpU,Hq, n P N, of pFtqtPr0,T s-elementary stochastic processes suchthat for all ε P p0,8q it holds that
limnÑ8
Pˆż T
0
Xs ´Xns
2HSpQ12pUq,Hqq ds ě ε
˙
“ 0. (6.59)
Lemma 6.2.25 follows, e.g., from Proposition 2.3.8 in Prevot & Rockner [24].In the next result, Theorem 6.2.26, the existence and uniqueness of the stochasticintegral is established (cf., e.g., Proposition 2.26 in Karatzas & Shreve [16]).
Theorem 6.2.26 (Stochastic integral). Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq beseparable R-Hilbert spaces, let T P p0,8q, let Q P L1pUq be nonnegative and sym-metric, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, and let W : r0, T s ˆΩ Ñ U be astandard Q-Wiener process w.r.t. pFtqtPr0,T s. Then there exists a unique linear map-
ping I :
X P L0`
PP,pFtqtPr0,T s ; ¨HSpQ12pUq,Hq
˘
: P`
∫T0 Xs2HSpQ12pUq,Hq
ds ă 8˘
“
1(
Ñ L0pP ; ¨Hq which satisfies
(i) that for all Xn P DpIq, n P N, with limnÑ8E“
mint1,şT
0Xn
s 2HSpQ12pUq,Hq
dsu‰
“
0 it holds that limnÑ8E“
mint1, IpXnqHu‰
“ 0 (continuity) and
(ii) that for all s P r0, T s, t P ps, T s and all Fs/BpLpU,Hqq-simple X : Ω Ñ LpU,Hqit holds P -a.s. that
Ip1ps,tsXq “ X pWt ´Wsq (6.60)
(stochastic integration of elementary stochastic processes).
170 CHAPTER 6. STOCHASTIC PROCESSES
Definition 6.2.27 (Stochastic integral on the entire time interval). Let T P p0,8q,let pΩ,F , P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uqbe separable R-Hilbert spaces, let Q P L1pUq be nonnegative and symmetric, andlet W : r0, T s ˆ Ω Ñ U be a standard Q-Wiener process w.r.t. pFtqtPr0,T s. Then we
denote by IW :
X P L0`
PP,pFtqtPr0,T s ; ¨HSpQ12pUq,Hq
˘
: P`
∫T0 Xs2HSpQ12pUq,Hq
ds ă
8˘
“ 1(
Ñ L0pP ; ¨Hq the unique linear mapping which satisfies
(i) that for all Xn P DpIW q, n P N, with limnÑ8E“
mint1,şT
0Xn
s 2HSpQ12pUq,Hq
dsu‰
“ 0 it holds that limnÑ8E“
mint1, IW pXnqHu‰
“ 0 (continuity) and
(ii) that for all s P r0, T s, t P ps, T s and all Fs/BpLpU,Hqq-simple X : Ω Ñ LpU,Hqit holds P -a.s. that
IW p1ps,tsXq “ X pWt ´Wsq (6.61)
(stochastic integration of elementary stochastic processes).
Question 6.2.28. Let T P p0,8q, a, b P r0, T s with a ă b, let pΩ,F , P, pFtqtPr0,T sqbe a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be separable R-Hilbertspaces, let Q P L1pUq be nonnegative and symmetric, let W : r0, T sˆΩ Ñ U be a stan-dard Q-Wiener process w.r.t. pFtqtPr0,T s, and let X P L0
`
PP,pFtqtPr0,T s ; ¨HSpQ12pUq,Hq
˘
.
Is it true or is it not true that r0, T s ˆ Ω ÞÑ 1pa,bqpsq ¨ Xspωq P HSpQ12pUq, Hqq P
L0`
PP,pFtqtPr0,T s ; ¨HSpQ12pUq,Hq
˘
?
Definition 6.2.29 (Stochastic integral). Let T P p0,8q, a, b P r0, T s with a ă b,let pΩ,F , P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uqbe separable R-Hilbert spaces, let Q P L1pUq be nonnegative and symmetric, letW : r0, T s ˆ Ω Ñ U be a standard Q-Wiener process w.r.t. pFtqtPr0,T s, and let X P
L0`
PP,pFtqtPr0,T s ; ¨HSpQ12pUq,Hq
˘
satisfy that it holds P-a.s. that ∫ ba Xs2HSpQ12pUq,Hq
ds
ă 8. Then we denote byşb
aXs dWs P L
0pP; ¨Hq the element given byşb
aXs dWs “
IW p1pa,bqXq.
Exercise 6.2.30. Let T P r0,8q, a, b P r0, T s with a ă b, let pΩ,F ,P, pFtqtPr0,T sqbe a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be separable R-Hilbertspaces, let Q P L1pUq be nonnegative and symmetric, let W : r0, T sˆΩ Ñ U be a stan-dard Q-Wiener process w.r.t. pFtqtPr0,T s, and let X P L0
`
PpP,pFtqtPr0,T sq; ¨HSpQ12pUq,Hq
˘
be such thatşb
aE“
Xs2HSpQ12pUq,Hq
‰
ds ă 8. Prove thatşb
aE“
Xs2H
‰
ds ă 8 and
E“ şb
aXs dWs
‰
“ 0.
6.2. STOCHASTIC INTEGRATION 171
In the following a few properties of the stochastic integral are collected. For thisthe following lemma is used.
Lemma 6.2.31. Let T P r0,8q, t P r0, T s, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pS,Sq be a measurable space, let X : Ω Ñ S be an F/S-measurable mappingand let Y : Ω Ñ S be an Ft/S-measurable mapping such that it holds P-a.s. thatX “ Y . Then it holds that X is Ft/S-measurable.
Proof of Lemma 6.2.31. First, note that the assumption that X “ Y P-a.s. showsthat there exists a measurable set A P F with the property that PpAq “ 1 and withthe property that for all ω P A it holds that Xpωq “ Y pωq. Next observe that for allB P S it holds that
X´1pBq “
“
X´1pBq X A
‰
Y“
X´1pBqzA
‰
“ tω P A : Xpωq P Bu Y“
X´1pBqzA
‰
“ tω P A : Y pωq P Bu Y“
X´1pBqzA
‰
““
Y ´1pBq X A
‰
Y“
X´1pBqzA
‰
.
(6.62)
Moreover, observe that the assumption that pFtqtPr0,T s is a normal filtration togetherwith the fact that PpAq “ 1 implies that
A,Ac P F0 Ď Ft Ď F . (6.63)
This and the assumption that Y is Ft/S-measurable prove that for all B P S it holdsthat
Y ´1pBq X A P Ft. (6.64)
Furthemore, note that the monotonicity of the probability measure P ensures thatfor all B P S it holds that PpX´1pBqzAq “ 0. The assumption that pFtqtPr0,T s isnormal hence shows that for all B P S it holds that
X´1pBqzA P Ft. (6.65)
Combining (6.62) with (6.64) and (6.65) proves that for all B P S it holds thatX´1pBq P Ft. The proof of Lemma 6.2.31 is thus completed.
Consider the setting of Lemma 6.2.31 and let pV, ¨V q be a separable Banachspace. Then Lemma 6.2.31, in particular, proves that for all t1, t2 P r0, T s witht1 ď t2 it holds that
L0`
P|Ft1 ; ¨V˘
Ď L0`
P|Ft2 ; ¨V˘
Ď L0`
P; ¨V˘
. (6.66)
172 CHAPTER 6. STOCHASTIC PROCESSES
In Lemma 6.2.31 it is crucial that the filtration is normal. Let us collect a few prop-erties of the stochastic integral with possibly infinite dimensional Wiener processesas integrator processes.
6.2. STOCHASTIC INTEGRATION 173
Theorem 6.2.32 (Properties of the stochastic integral). Let T P p0,8q, a, b P r0, T swith a ă b, let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be separable R-Hilbert spaces,let Q P L1pUq be nonnegative and symmetric, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let W : r0, T s ˆ Ω Ñ U be a standard Q-Wiener process w.r.t. pFtqtPr0,T s, andlet X : r0, T sˆΩ Ñ HSpQ12pUq, Hq be an pFtqtPr0,T s-predictable stochastic processes
with the property that it holds P-a.s. thatşb
aXs
2HSpQ12pUq,Hq ds ă 8. Then
(i) it holds thatşb
aXs dWs P L
0`
P|Fb ; ¨H˘
, i.e., it holds thatşb
aXs dWs is Fb/BpHq-
measurable,
(ii) it holds that pşt
aXs dWsqtPra,bs is an pFtqtPra,bs-adapted stochastic process,
(iii) for all α, β P R and all pFtqtPr0,T s-predictable stochastic processes Y, Z : r0, T sˆ
Ω Ñ HSpQ12pUq, Hq with P` şb
aYs
2HSpQ12pUq,Hq
`Zs2HSpQ12pUq,Hq
ds ă 8˘
“
1 it holds P-a.s. thatż b
a
rαYs ` βZss dWs “ α
ż b
a
Ys dWs ` β
ż b
a
Zs dWs, (6.67)
(iv) for all pFtqtPr0,T s-predictable stochastic processes Y : r0, T sˆΩ Ñ HSpQ12pUq, Hq
withşb
aE“
Ys2HSpQ12pUq,Hq
‰
ds ă 8 it holds that
E
«
›
›
›
›
ż b
a
Ys dWs
›
›
›
›
2
H
ff
“
ż b
a
E
”
Ys2HSpQ12pUq,Hq
ı
ds ă 8, (Ito’s isometry)
›
›
›
›
ż b
a
Ys dWs
›
›
›
›
L2pP;¨Hq
“
ˆż b
a
Ys2L2pP;¨
HSpQ12pUq,Hqqds
˙
12
ă 8, (6.68)
E
„ż b
a
Ys dWs
“ 0, (6.69)
(v) for all p P r2,8q it holds that
›
›
›
›
ż b
a
Xs dWs
›
›
›
›
LppP;¨Hq
ď
c
p pp´ 1q
2
ˆż b
a
Xs2LppP;¨
HSpQ12pUq,Hqqds
˙
12
˜
E
«
›
›
›
›
ż b
a
Xs dWs
›
›
›
›
p
H
ff¸1p
ď
c
p pp´ 1q
2
ˆż b
a
´
E
”
Xsp
HSpQ12pUq,Hq
ı¯2pds
˙
12
,
(Burkholder-Davis-Gundy inequality I)
174 CHAPTER 6. STOCHASTIC PROCESSES
(vi) there exists an up to indistinguishability unique pFtqtPra,bs-adapted stochasticprocess V : ra, bs ˆ Ω Ñ H with continuous sample paths which satisfies thatfor all t P ra, bs it holds P -a.s. that Vt “
şt
aXs dWs (V is called a continuous
modification of pşt
aXs dWsqtPra,bs),
(vii) and for all continuous modifications V : ra, bsˆΩ Ñ H of pşt
aXs dWsqtPra,bs and
all p P r2,8q it holds that›
›
›
›
›
supsPra,bs
VsH
›
›
›
›
›
LppP;|¨|Rq
ď p
ˆż b
a
Xs2LppP;¨
HSpQ12pUq,Hqqds
˙
12
,
˜
E
«
supsPra,bs
VspH
ff¸1p
ď p
ˆż b
a
´
E
”
Xsp
HSpQ12pUq,Hq
ı¯2p
ds
˙
12
.
(Burkholder-Davis-Gundy inequality II)
The statements of Theorem 6.2.32 and their proofs can, for example, be found in[24] and [7].
Exercise 6.2.33 (Stochastic integration of L2-continuous stochastic processes). LetT P p0,8q, d,m P N, let pΩ,F , P, pFtqtPr0,T sq be a stochastic basis, let W : r0, T sˆΩ ÑRm be an m-dimensional standard pFtqtPr0,T s-Brownian motion, let a, b P r0, T s witha ď b and let X : r0, T s ˆ Ω Ñ Rdˆm be an pFtqtPr0,T s-predictable stochastic processwith X P Cpr0, T s, L2pP ; ¨
Rdˆmqq. Prove then that
ż b
a
Xs dWs “ L2pP ; ¨
Rdq´ lim
nÑ8
«
n´1ÿ
k“0
Xpa` kpb´aq
nq
´
Wa` pk`1qpb´aq
n´W
a` kpb´aqn
¯
ff
.
(6.70)
6.2.6 Elementary processes revisited
In the literature (see, e.g., Definition 2.3.1 in [24]) a slightly different notion ofan elementary stochastic process is often given. In Proposition 6.2.37 below weshow that these different definitions (cf. Definition 2.3.1 in [24] and Definition 6.2.20above) are equivalent. Our proof of Proposition 6.2.37 uses Exercise 6.2.35 below.Exercise 6.2.35 below can, e.g., be proved by using the following lemma.
Lemma 6.2.34. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space andlet ϕ P CpH,Kq, v P Hzt0u with the property that for all u P H it holds thatϕpuq ¨ 〈v, u〉H “ 0. Then it holds for all u P H that ϕpuq “ 0.
6.2. STOCHASTIC INTEGRATION 175
Proof of Lemma 6.2.34. Observe that for all λ P Kzt0u, w P rspantvusJ it holds that
0 “ ϕpλv ` wq ¨ 〈v, λv ` w〉H “ ϕpλv ` wq ¨ λ ¨ v2Hlooomooon
‰0
. (6.71)
This implies that for all for all λ P Kzt0u, w P rspantvusJ it holds that
ϕpλv ` wq “ 0. (6.72)
The fact that the set!
λv ` w P H : λ P Kzt0u, w P rspantvusJ)
(6.73)
is dense in H together with the assumption that ϕ is continuous hence implies that forall u P H it holds that ϕpuq “ 0. The proof of Lemma 6.2.34 is thus completed.
Exercise 6.2.35. Let K P tR,Cu, n P t2, 3, . . . u, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2ube K-Hilbert spaces and let A1, . . . , An P LpH1, H2q with the property that for all u PH1 it holds that A1u P tA2u,A3u, . . . , Anuu. Prove then that A1 P tA2, A3, . . . , Anu.
Exercise 6.2.35 can, e.g., be proved by using Lemma 6.2.34. In our proof ofProposition 6.2.37 below we also use the following exercise.
Exercise 6.2.36. Let pΩ1,F1q be a measurable space, let Ω2 be a set and let f : Ω1 Ñ
Ω2 be a mapping with the property that the set impfq is finite. Then f is F1/PpΩ2q-measurable if and only if for all ω P impfq it holds that f´1ptωuq P F1.
Proposition 6.2.37 (Uniform and strong measurability). Let K P tR,Cu, let pΩ,Fqbe a measurable space, let pHk, 〈¨, ¨〉Hk , ¨Hkq, k P t1, 2u, be K-Hilbert spaces and letY : Ω Ñ LpH1, H2q be a function with the property that impY q is a finite set. Thenit holds that Y is F/BpLpH1, H2qq-measurable if and only if for all v P H1 it holdsthat Ω Q ω ÞÑ Y pωqv P H2 is F/BpH2q-measurable.
Proof of Proposition 6.2.37. It is clear that if Y is F/BpLpH1, H2qq-measurable, thenit holds for all v P H1 that Y v is F/BpH2q-measurable. We thus assume in thefollowing that for all v P H1 it holds that Y v is F/BpH2q-measurable. Let A1 P impY qbe arbitrary. We now prove that Y ´1ptA1uq P F . This and Exercise 6.2.36 will thenshow that Y is F/BpLpH1, H2qq-measurable. W.l.o.g. we assume that impY qztA1u ‰
H. As impY q is a finite set, there exists n P t2, 3, . . . u and A2, . . . , An P impY qztA1u
such thattA1, A2, . . . , Anu “ impY q. (6.74)
176 CHAPTER 6. STOCHASTIC PROCESSES
Exercise 6.2.35 implies that there exists a vector u P H1 such that
A1u R tA2u, . . . , Anuu . (6.75)
This and the assumption that Y u is F/BpH2q-measurable imply that
F Q pY uq´1ptA1uuq “ Y ´1
ptA1uq. (6.76)
Exercise 6.2.36 thus completes the proof of Proposition 6.2.37.
Exercise 6.2.38. Let pΩ,Fq be a measurable space, let pE, dEq be a metric spaceand let f : Ω Ñ E be a mapping with the property that the set impfq is finite. Thenf is F/PpEq-measurable if and only if f is F/BpEq-measurable.
6.2.7 Cylindrical Wiener process
The following presentations are based on [24] and Chapter 5 in [14]. Let pU, 〈¨, ¨〉U , ¨Uqbe an R-Hilbert space and let Q P L1pUq be nonnegative and symmetric. In Subsec-tion 6.1.1 the notion of a standard Q-Wiener process is presented. The covarianceoperator Q associated to a standard Q-Wiener process is a nonnegative, symmetric,and nuclear linear operator on the Hilbert space U on which the standard Wienerprocess takes values in. In many situations one is interested in an infinite dimen-sional Wiener process with a covariance operator that is a nonnegative and symmetricbounded linear operator which is not a nuclear linear operator (such as, for example,the identity operator on an infinite dimensional Hilbert space). This can be achievedby the concept of a cylindrical Wiener process.
Definition 6.2.39 (Cylindrical Wiener process). Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hqand pH1, 〈¨, ¨〉H1
, ¨H1q be R-Hilbert spaces with H Ď H1 continuously, let Q P LpHq
and Q1 P L1pH1q be nonnegative and symmetric with the property that
`
Q12pHq,
›
›Q´12p¨q›
›
H
˘
“`
Q12
1 pH1q,›
›Q´12
1 p¨q›
›
H1
˘
, (6.77)
let pΩ,F ,P, pFtqtPr0,T sq be a filtered probability space, and let W : r0, T s ˆ Ω Ñ H1
be a standard Q1-Wiener process w.r.t. pFtqtPr0,T s. Then W is called a cylindricalQ-Wiener process w.r.t. pFtqtPr0,T s.
Let T P r0,8q, let pH, 〈¨, ¨〉H , ¨Hq be an R-Hilbert space, let Q P LpHq benonnegative and symmetric, let pΩ,F ,P, pFtqtPr0,T sq be a filtered probability space,and let pWtqtPr0,T s be a cylindrical Q-Wiener process w.r.t. pFtqtPr0,T s. The cylindricalQ-Wiener process pWtqtPr0,T s thus, in general, does not take values in the Hilbert space
6.2. STOCHASTIC INTEGRATION 177
H, on which the covariance operator Q associated to W is defined, but on a largerHilbert space with a weaker topology into which H is continuously embedded. Moreresults on cylindrical Q-Wiener processes can be found in Section 2.5 in Prevot &Rockner [24].
178 CHAPTER 6. STOCHASTIC PROCESSES
Part III
Stochastic Partial DifferentialEquations (SPDEs)
179
Chapter 7
Solutions of SPDEs
7.1 Existence, uniqueness and properties of mild
solutions of SPDEs
7.1.1 Mild solutions of SPDEs
The next definition presents what we mean by a stochastic partial differential equa-tion and a mild solution of it.
181
182 CHAPTER 7. SOLUTIONS OF SPDES
Definition 7.1.1 (Mild solutions). Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be sep-arable R-Hilbert spaces, let Q P LpUq be nonnegative and symmetric, let A : DpAq ĎH Ñ H be a symmetric diagonal linear operator with suppσP pAqq ă 8, let η PpsuppσP pAqq,8q, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces
associated to η ´ A, let T P r0,8q, α, β, γ P R, O P BpHγq, F PM`
BpOq,BpHαq˘
,
B PM`
BpOq,BpHSpQ12pUq, Hβqq˘
, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, letξ PMpF0,BpOqq, let pWtqtPr0,T s be a cylindrical Q-Wiener process w.r.t. pFtqtPr0,T s,and let X : r0, T s ˆ Ω Ñ O be an pFtqtPr0,T s-predictable stochastic process with theproperty that for all t P r0, T s it holds P-a.s. that
ż t
0
eApt´sqF pXsqHγ ` eApt´sqBpXsq
2HSpQ12pUq,Hγq
ds ă 8 (7.1)
and Xt “ eAtξ `
ż t
0
eApt´sqF pXsq ds`
ż t
0
eApt´sqBpXsq dWs. (7.2)
Then X is said to be a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.3)
Equation (7.3) is referred to as stochastic partial differential equation (SPDE), thefunction F is the nonlinear part of the drift coefficient function Av ` F pvq, v P O,and the function B is the diffusion coefficient function of the SPDE (7.3).
7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 183
Example 7.1.2 (Ornstein-Uhlenbeck processes (stochastic heat equation with ad-ditive noise)). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplaceoperator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq (see Definition 3.5.7),let ξ P L2pBorelp0,1q; |¨|Rq, let en P L2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all
n P N and Borelp0,1q-almost all x P p0, 1q it holds that enpxq “?
2 sinpnπxq, letpΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let rn P r0,8q, n P N, be a bounded se-quence of nonnegative real numbers, let Q : L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rqbe the mapping with the property that for all v P L2pBorelp0,1q; |¨|Rq it holds that
Qv “8ÿ
n“1
rn 〈en, v〉L2pBorelp0,1q;|¨|Rqen, (7.4)
let pWtqtPr0,T s be a cylindrical Q-Wiener process w.r.t. pFtqtPr0,T s, and let X : r0, T s ˆΩ Ñ H be an pFtqtPr0,T s-predictable stochastic process which fulfills that for all t Pr0, T s it holds P-a.s. that
Xt “ eAtξ `
ż t
0
eApt´sq dWs. (7.5)
Then X is a mild solution of the SPDE
dXt “ AXt dt` dWt, t P r0, T s, X0 “ ξ. (7.6)
Sometimes one also writes
dXtpxq “B2
Bx2Xtpxq dt` dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ ξpxq (7.7)
for t P r0, T s, x P p0, 1q as a short form for (7.6).
In the following we investigate a few further properties of mild solutions of SPDEs.To this end we frequently use the following setting.
7.1.2 A setting for SPDEs with globally Lipschitz continuousnonlinearities
Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be two separable R-Hilbert spaces, letA : DpAq Ď H Ñ H be a symmetric diagonal linear operator with suppσP pAqq ă0, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associatedto ´A, let T P r0,8q, p P r2,8q, γ P R, η P r0, 1q, β P rγ ´ η2, γs, F P
C0,1pHγ, Hγ´ηq, B P C0,1pHγ, HSpU,Hβqq, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let ξ P LppP|F0 ; ¨Hγ q, and let pWtqtPr0,T s be a cylindrical IdU -Wiener process
w.r.t. pFtqtPr0,T s.
184 CHAPTER 7. SOLUTIONS OF SPDES
7.1.3 A strong perturbation estimate for SPDEs
The following estimate is a special case of an inequality known as Minkowski’s integralinequality; see, for instance, Apendix A.1 in Stein [26] and Theorem 202 in Hardy,Littlewood & Polya [9].
Proposition 7.1.3. Let T P r0,8q, p P r1,8q, let pΩ,F ,Pq be a probability space,and let Y : r0, T s ˆ Ω Ñ r0,8s be a product measurable stochastic process. Then
ˆ
E„ˇ
ˇ
ˇ
ˇ
ż T
0
Ys ds
ˇ
ˇ
ˇ
ˇ
p˙1p
ď
ż T
0
´
E“
|Ys|p‰
¯1pds. (7.8)
In the next result, Proposition 7.1.4, we establish, in the setting of Section 7.1.2,a certain strong perturbation result for two arbitrary predictable stochastic processesX1 and X2 satisfying supsPr0,T s maxkPt1,2u X
ks LppP;¨Hγ q
ă 8. In particular, we em-
phasize in the setting of Section 7.1.2 that neither X1 nor X2 in Proposition 7.1.4need to be mild solutions of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.9)
In the statement of Proposition 7.1.4 and in a number of other results in this andthe next chapter we use the functions Er : r0,8q Ñ r0,8q, r P p0,8q, introduced inDefinition 1.3.1 in Chapter 1 above.
Proposition 7.1.4 (Perturbation estimate). Assume the setting in Section 7.1.2and let X1, X2 : r0, T s ˆ Ω Ñ Hγ be pFtqtPr0,T s-predictable stochastic processes withthe property that supsPr0,T s maxkPt1,2u X
ks LppP;¨Hγ q
ă 8. Then
suptPr0,T s
›
›X1t ´X
2t
›
›
LppP;¨Hγ q
ď Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qq
¨?
2 suptPr0,T s
›
›
›
›
„
X1t ´
ż t
0
eApt´sqF pX1s q ds´
ż t
0
eApt´sqBpX1s q dWs
`
„ż t
0
eApt´sqF pX2s q ds`
ż t
0
eApt´sqBpX2s q dWs ´X
2t
›
›
›
›
LppP;¨Hγ q
ă 8.
(7.10)
7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 185
Proof of Proposition 7.1.4. Note that the Minkowski inequality ensures that for allt P r0, T s it holds that
›
›X1t ´X
2t
›
›
LppP;¨Hγ q
ď
›
›
›
›
›
„
X1t ´
ˆż t
0
eApt´sqF pX1s q ds`
ż t
0
eApt´sqBpX1s q dWs
˙
`
„ˆż t
0
eApt´sqF pX2s q ds`
ż t
0
eApt´sqBpX2s q dWs
˙
´X2t
›
›
›
›
›
LppP;¨Hγ q
`
›
›
›
›
›
ˆż t
0
eApt´sqF pX1s q ds`
ż t
0
eApt´sqBpX1s q dWs
˙
´
ˆż t
0
eApt´sqF pX2s q ds`
ż t
0
eApt´sqBpX2s q dWs
˙
›
›
›
›
›
LppP;¨Hγ q
.
(7.11)
Again the Minkowski inequality hence implies that for all t P r0, T s it holds that
›
›X1t ´X
2t
›
›
LppP;¨Hγ q
ď
›
›
›
›
›
„
X1t ´
ˆż t
0
eApt´sqF pX1s q ds`
ż t
0
eApt´sqBpX1s q dWs
˙
`
„ˆż t
0
eApt´sqF pX2s q ds`
ż t
0
eApt´sqBpX2s q dWs
˙
´X2t
›
›
›
›
›
LppP;¨Hγ q
`
›
›
›
›
ż t
0
eApt´sq“
F pX1s q ´ F pX
2s q‰
ds
›
›
›
›
LppP;¨Hγ q
`
›
›
›
›
ż t
0
eApt´sq“
BpX1s q ´BpX
2s q‰
dWs
›
›
›
›
LppP;¨Hγ q
.
(7.12)
186 CHAPTER 7. SOLUTIONS OF SPDES
Next note that Holder’s inequality implies that for all t P r0, T s it holds that
›
›
›
›
ż t
0
eApt´sq“
F pX1s q ´ F pX
2s q‰
ds
›
›
›
›
LppP;¨Hγ q
ď
ż t
0
›
›eApt´sq“
F pX1s q ´ F pX
2s q‰›
›
LppP;¨Hγ qds
ď
ż t
0
F pX1s q ´ F pX
2s qLppP;¨Hγ´η
q
pt´ sqηds
ď |F |C0,1pHγ ,Hγ´ηq
ż t
0
X1s ´X
2s LppP;¨Hγ q
pt´ sqηds
ď |F |C0,1pHγ ,Hγ´ηq
g
f
f
e
ż t
0
pt´ sq´η ds
ż t
0
X1s ´X
2s
2LppP;¨Hγ q
pt´ sqηds
“ |F |C0,1pHγ ,Hγ´ηq
d
tp1´ηq
p1´ ηq
ż t
0
pt´ sq´η X1s ´X
2s
2LppP;¨Hγ q
ds.
(7.13)
Furthermore, observe that for all t P r0, T s it holds that
›
›
›
›
ż t
0
eApt´sq“
BpX1s q ´BpX
2s q‰
dWs
›
›
›
›
LppP;¨Hγ q
ď
d
p pp´1q2
ż t
0
eApt´sq rBpX1s q ´BpX
2s qs
2LppP;¨HSpU,Hγ qq
ds
ď
d
p pp´1q2
ż t
0
pt´ sq´η BpX1s q ´BpX
2s q
2LppP;¨HSpU,Hγ´η2q
qds
ď |B|C0,1pHγ ,HSpU,Hγ´η2qq
d
p pp´1q2
ż t
0
pt´ sq´η X1s ´X
2s
2LppP;¨Hγ q
ds.
(7.14)
7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 187
Combining (7.13) and (7.14) proves that for all t P r0, T s it holds that
›
›
›
›
ż t
0
eApt´sq“
F pX1s q ´ F pX
2s q‰
ds
›
›
›
›
LppP;¨Hγ q
`
›
›
›
›
ż t
0
eApt´sq“
BpX1s q ´BpX
2s q‰
dWs
›
›
›
›
LppP;¨Hγ q
ď
„
|F |C0,1pHγ ,Hγ´ηqtp1´ηq2?
1´η` |B|C0,1pHγ ,HSpU,Hγ´η2qq
?p pp´1q?
2
¨
g
f
f
e
ż t
0
X1s ´X
2s
2LppP;¨Hγ q
pt´ sqηds.
(7.15)
Putting this into (7.12) and using the fact that @ a, b P R : pa`bq2 ď 2a2`2b2 provesthat for all t P r0, T s it holds that
›
›X1t ´X
2t
›
›
2
LppP;¨Hγ q
ď 2
›
›
›
›
›
„
X1t ´
ˆż t
0
eApt´sqF pX1s q ds`
ż t
0
eApt´sqBpX1s q dWs
˙
`
„ˆż t
0
eApt´sqF pX2s q ds`
ż t
0
eApt´sqBpX2s q dWs
˙
´X2t
›
›
›
›
›
2
LppP;¨Hγ q
`
”
|F |C0,1pHγ ,Hγ´ηq
?2T
p1´ηq2?
1´η` |B|C0,1pHγ ,HSpU,Hγ´η2qq
a
p pp´ 1qı2
¨
ż t
0
X1s ´X
2s
2LppP;¨Hγ q
pt´ sqηds.
(7.16)
The generalized Gronwall inequality in Corollary 1.4.6 hence completes the proof ofProposition 7.1.4.
The next corollary, Corollary 7.1.5, is an immediate consequence of the strongperturbation estimate in Proposition 7.1.4 above.
188 CHAPTER 7. SOLUTIONS OF SPDES
Corollary 7.1.5 (Perturbation in the initial value). Assume the setting in Sec-tion 7.1.2 and let X1, X2 : r0, T s ˆ Ω Ñ Hγ be pFtqtPr0,T s-predictable stochastic pro-cesses with the property that supsPr0,T s maxkPt1,2u X
ks LppP;¨Hγ q
ă 8 and with the
property that for all t P r0, T s, k P t1, 2u it holds P-a.s. that
Xkt “ eAtXk
0 `
ż t
0
eApt´sqF pXks q ds`
ż t
0
eApt´sqBpXks q dWs. (7.17)
Then
suptPr0,T s
›
›X1t ´X
2t
›
›
LppP;¨Hγ qď?
2›
›X10 ´X
20
›
›
LppP;¨Hγ q
¨ Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qq
ă 8.(7.18)
7.1.4 Uniqueness of mild solutions of SPDEs
7.1.4.1 Uniqueness of predictable mild solutions of SEEs with globallyLipschitz continuous coefficients
As an immediate consequence of Corollary 7.1.5, we obtain, under suitable assump-tions, uniqueness of mild solutions of SPDEs; cf., e.g., Theorem 7.4 (i) in Da Prato& Zabczyk [7].
Corollary 7.1.6. Assume the setting in Subsection 7.1.2, let X1, X2 : r0, T s ˆ Ω ÑHγ be mild solutions of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (7.19)
and assume that maxkPt1,2u suptPr0,T s Xkt LppP;¨Hγ q
ă 8. Then it holds that X1 and
X2 are modifications of each other, i.e., it holds for all t P r0, T s that P`
X1t “ X2
t
˘
“
1.
7.1.4.2 Uniqueness of left-continuous mild solutions of SEEs with semi-globally Lipschitz continuous coefficients
The proof of the next result, Proposition 7.1.7, is similiar to the proof of Theorem 7.4in Da Prato & Zabczyk [7]. See also, e.g., Lemma 8.2 in [28] for the next result.
7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 189
Proposition 7.1.7 (Local solutions). Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq beseparable R-Hilbert spaces, let A : DpAq Ď H Ñ H be a symmetric diagonal lin-ear operator with suppσP pAqq ă 0, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family ofinterpolation spaces associated to ´A, let T P p0,8q, γ P R, η P r0, 1q, F P
CpHγ, Hγ´ηq, B P CpHγ, HSpU,Hγ´η2qq satisfy for all bounded sets E Ď Hγ that|F |E|C0,1pE,Hγ´ηq`|B|E|C0,1pE,HSpU,Hγ´η2qq ă 8, let pΩ,F ,Pq be a probability space witha normal filtration pFtqtPr0,T s, let τk : Ω Ñ r0, T s, k P t1, 2u, be pFtqtPr0,T s-stoppingtimes, and let Xk : r0, T s ˆ Ω Ñ Hγ, k P t1, 2u, be pFtqtPr0,T s-adapted stochas-tic processes with left-continuous and bounded sample paths and with the propertythat for all k P t1, 2u, t P r0, T s it holds P-a.s. that
şt
01tsăτku
“
ept´sqAF pXks qHγ `
ept´sqABpXks q
2HSpU,Hγq
‰
ds ă 8 and
Xkt 1ttďτku (7.20)
“
„
etAXk0 `
ż t
0
1tsăτku ept´sqAF pXk
s q ds`
ż t
0
1tsăτku ept´sqABpXk
s q dWs
1ttďτku.
Then P`
@ t P r0, T s : 1tX10“X
20uX1
mintt,τ1,τ2u“ 1tX1
0“X20uX2
mintt,τ1,τ2u
˘
“ 1.
Proof of Proposition 7.1.7. Throughout this proof let Ω P F be the set given byΩ “ tX0 “ Y0u, let %r,k : Ω Ñ r0, T s, r P p0,8q, k P t1, 2u, be the mappings with theproperty that for all r P p0,8q, k P t1, 2u it holds that
%r,k “ inf`
tT u Y tt P r0, T s : Xkt Hγ ą ru
˘
, (7.21)
let ρr : Ω Ñ r0, T s, r P p0,8q, be the mappings with the property that for allr P p0,8q it holds that ρr “ mint%r,1, %r,2, τ1, τ2u, and let Xk,r : r0, T s ˆ Ω Ñ Hγ,k P t1, 2u, r P p0,8q, be the mappings with the property that for all k P t1, 2u,r P p0,8q, t P r0, T s it holds that Xk,r
t “ 1ΩXttďρruXkt . Note for all r P p0,8q,
k P N that %r,k and ρr are pFtqtPr0,T s-stopping times. This ensures that for everyr P p0,8q, k P N it holds that Xk,r is a pFtqtPr0,T s-predictable stochastic process withleft-continuous sample paths. Moreover, observe that for all r P p0,8q, t P r0, T s it
190 CHAPTER 7. SOLUTIONS OF SPDES
holds P-a.s. that
X1,rt ´X2,r
t “ 1ΩXttďρru
ż t
0
ept´sqA“
1tsăτ1uF pX1s q ´ 1tsăτ2uF pX
2s q‰
ds
` 1ΩXttďρru
ż t
0
ept´sqA“
1tsăτ1uBpX1s q ´ 1tsăτ2uBpX
2s q‰
dWs
“ 1ΩXttďρru
ż t
0
1ΩXtsăρru ept´sqA
“
F pX1s q ´ F pX
2s q‰
ds
` 1ΩXttďρru
ż t
0
1ΩXtsăρru ept´sqA
“
BpX1s q ´BpX
2s q‰
dWs
“ 1ΩXttďρru
ż t
0
1ΩXtsăρru ept´sqA
“
F pX1,rs q ´ F pX
2,rs q
‰
ds
` 1ΩXttďρru
ż t
0
1ΩXtsăρru ept´sqA
“
BpX1,rs q ´BpX
2,rs q
‰
dWs.
(7.22)
This implies for all r P p0,8q that
supsPr0,T s
E“
X1,rs ´X2,r
s 2Hγ
‰
“ supsPr0,T s
E“
1ΩXtsďρruX1s ´X
2s
2Hγ
‰
ď supsPr0,T s
E“
1ΩXtsďρruXtρr“0uX1s ´X
2s
2Hγ
‰
` supsPr0,T s
E“
1ΩXtsďρruXtρrą0uX1s ´X
2s
2Hγ
‰
“ supsPr0,T s
E“
1ΩXtsďρruXtρrą0uX1s ´X
2s
2Hγ
‰
ď supsPr0,T s
E“
1ΩXtρrą0uX1mints,ρru ´X
2mints,ρru
2Hγ
‰
ď 2 ¨ supsPr0,T s
E“
1tρrą0uX1mints,ρru
2Hγ ` 1tρrą0uX
2mints,ρru
2Hγ
‰
ď 4r2ă 8.
(7.23)
Moreover, equation (7.22) ensures that for all r P p0,8q, t P r0, T s it holds that
E“
X1,rt ´X2,r
t 2Hγ
‰
ď 2E
«
ˇ
ˇ
ˇ
ˇ
ż t
0
1ΩXtsăρru›
›ept´sqA“
F pX1,rs q ´ F pX
2,rs q
‰›
›
Hγds
ˇ
ˇ
ˇ
ˇ
2ff
` 2E
«
›
›
›
›
ż t
0
1ΩXtsăρru ept´sqA
“
BpX1,rs q ´BpX
2,rs q
‰
dWs
›
›
›
›
2
Hγ
ff
.
(7.24)
7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 191
Ito’s isometry hence proves that for all r P p0,8q, t P r0, T s it holds that
E“
X1,rt ´X2,r
t 2Hγ
‰
ď 2E
«
ˇ
ˇ
ˇ
ˇ
ż t
0
1ΩXtsăρru pt´ sq´ηF pX1,r
s q ´ F pX2,rs qHγ´η ds
ˇ
ˇ
ˇ
ˇ
2ff
` 2
ż t
0
pt´ sq´η E“
1ΩXtsăρru BpX1,rs q ´BpX
2,rs q
2HSpU,Hγ´η2q
‰
ds.
(7.25)
This shows that for all r P p0,8q, t P r0, T s it holds that
E“
X1,rt ´X2,r
t 2Hγ
‰
ď 2
ˆż t
0
pt´ sq´η ds
˙ż t
0
pt´ sq´η E“
1tsăρru F pX1,rs q ´ F pX
2,rs q
2Hγ´η
‰
ds
` 2
ż t
0
pt´ sq´η E“
1tsăρru BpX1,rs q ´BpX
2,rs q
2HSpU,Hγ´η2q
‰
ds.
(7.26)
Hence, we obtain that for all r P p0,8q, t P r0, T s it holds that
E“
X1,rt ´X2,r
t 2Hγ
‰
ď2T p1´ηq
p1´ ηq
ż t
0
|F |txPHγ : xHγďru|2C0,1ptxPHγ : xHγďru,Hγ´ηq
pt´ sqηE“
X1,rs ´X2,r
s 2Hγ
‰
ds
` 2
ż t
0
|B|txPHγ : xHγďru|2C0,1ptxPHγ : xHγďru,HSpU,Hγ´η2qq
pt´ sqηE“
X1,rs ´X2,r
s 2Hγ
‰
ds.
(7.27)
Combining this with (7.23) allows us to apply Corollary 1.4.6 to obtain that for allt P r0, T s, r P p0,8q it holds that E
“
X1,rt ´ X2,r
t 2Hγ
‰
“ 0. Monotone convergencehence proves that for all t P r0, T s it holds that
E“
1ttďmintτ1,τ2uuX1t ´X
2t
2Hγ
‰
“ E
”
limrÑ8
1ttďρruX1t ´X
2t
2Hγ
ı
“ limrÑ8
E“
1ttďρruX1t ´X
2t
2Hγ
‰
“ limrÑ8
E“
X1,rt ´X2,r
t 2Hγ
‰
“ 0.(7.28)
This proves that for all t P r0, T s it holds P-a.s. that 1ttďmintτ1,τ2uu rX1t ´ X2
t s “ 0.Combining this with the fact that for every ω P Ω it holds that the function r0, T s Qt ÞÑ 1ttďmintτ1pωq,τ2pωquu rX
1t pωq ´X
2t pωqs P Hγ is left-continuous ensures that
P`
@ t P r0, T s : X1mintt,τ1,τ2u
“ X2mintt,τ1,τ2u
˘
“ P`
@ t P r0, T s : 1ttďmintτ1,τ2uu rX1t ´X
2t s “ 0
˘
“ P`
@ t P r0, T s XQ : 1ttďmintτ1,τ2uu rX1t ´X
2t s “ 0
˘
“ 1.
(7.29)
192 CHAPTER 7. SOLUTIONS OF SPDES
This completes the proof of Proposition 7.1.7.
Corollary 7.1.8 is an immediate consequence from Proposition 7.1.7.
Corollary 7.1.8. Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be separable R-Hilbertspaces, let T P p0,8q, γ P R, η P r0, 1q, F P CpHγ, Hγ´ηq, B P CpHγ, HSpU,Hγ´η2qq
satisfy for all bounded sets E Ď Hγ that |F |E|C0,1pE,Hγ´ηq ` |B|E|C0,1pE,HSpU,Hγ´η2qq ă
8, let pΩ,F ,Pq be a probability space with a normal filtration pFtqtPr0,T s, and letXk : r0, T s ˆ Ω Ñ Hγ, k P t1, 2u, be pFtqtPr0,T s-adapted stochastic processes withcontinuous sample paths such that for all k P t1, 2u, t P r0, T s it holds P-a.s. that
Xkt “ etAX1
0 `
ż t
0
ept´sqAF pXks q ds`
ż t
0
ept´sqABpXks q dWs. (7.30)
Then P`
@ t P r0, T s : X1t “ X2
t
˘
“ 1.
7.1.5 Existence and regularity of mild solutions of SPDEs
Theorem 7.1.9. Assume the setting in Subsection 7.1.2. Then there exists an upto modifications unique pFtqtPr0,T s-predictable stochastic process X : r0, T s ˆ Ω Ñ Hγ
which satisfies suptPr0,T s XtLppP;¨Hγ qă 8 and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.31)
Theorem 7.1.9 can be proved by a standard fixed point argument; see, e.g., Chap-ter 5 in [14].
7.1.6 A priori bounds for mild solutions of SPDEs
Definition 7.1.10. Let K P tR,Cu and let pV, ¨V q and pW, ¨W q be normed K-vector spaces. Then we denote by ¨LippV,W q : MpV,W q Ñ r0,8s the mapping with
the property that for all f PMpV,W q it holds that
fLippV,W q “ fp0qW ` |f |C0,1pV,W q . (7.32)
7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 193
Proposition 7.1.11. Assume the setting in Subsection 7.1.2 and let X : r0, T sˆΩ ÑHγ be the up to modifications unique pFtqtPr0,T s-predictable stochastic process whichsatisfies suptPr0,T s XtLppP;¨Hγ q
ă 8 and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.33)
Then
suptPr0,T s
XtLppP;¨Hγ qď?
2 max
1, ξLppP;¨Hγ q
(
¨ Ep1´ηq„
T 1´η?
2 F LippHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q BLippHγ ,HSpU,Hγ´η2qq
ă 8.(7.34)
Proof of Proposition 7.1.11. Observe that Theorem 6.2.32 implies that for all t Pr0, T s it holds that
XtLppP;¨Hγ qď X0LppP;¨Hγ q
`
ż t
0
›
›eApt´sq F pXsq›
›
LppP;¨Hγ qds
`
c
p pp´ 1q
2
„ż t
0
›
›eApt´sqBpXsq›
›
2
LppP;¨HSpU,Hγ qqds
12
ď X0LppP;¨Hγ q`
„
tp1´ηq
p1´ ηq
ż t
0
pt´ sq´η F pXsq2LppP;¨Hγ´η
qds
12
`
c
p pp´ 1q
2
„ż t
0
pt´ sq´η BpXsq2LppP;¨HSpU,Hγ´η2q
qds
12
.
(7.35)
This shows that for all t P r0, T s it holds that
XtLppP;¨Hγ qď X0LppP;¨Hγ q
`
„ż t
0
pt´ sq´η max!
1, Xs2LppP;¨Hγ q
)
ds
12
¨
«
F LippHγ ,Hγ´ηq
d
T p1´ηq
p1´ ηq` BLippHγ ,HSpU,Hγ´η2qqq
c
p pp´ 1q
2
ff
.
(7.36)
194 CHAPTER 7. SOLUTIONS OF SPDES
This proves that for all t P r0, T s it holds that
max!
1, Xt2LppP;¨Hγ q
)
ď 2 max!
1, X02LppP;¨Hγ q
)
`
ż t
0
pt´ sq´η max!
1, Xs2LppP;¨Hγ q
)
ds
¨
«
F LippHγ ,Hγ´ηq
d
2T p1´ηq
p1´ ηq` BLippHγ ,HSpU,Hγ´η2qq
a
p pp´ 1q
ff2
.
(7.37)
An application of Corollary 1.4.6 hence completes the proof of Proposition 7.1.11.
Definition 7.1.12. Let K P tR,Cu and let pV, ¨V q and pW, ¨W q be normed K-vector spaces. Then we denote by ¨LGpV,W q : MpV,W q Ñ r0,8s the mapping with
the property that for all f PMpV,W q it holds that
fLGpV,W q “ supvPV
„
fpvqWmaxt1, vV u
. (7.38)
Proposition 7.1.13. Assume the setting in Subsection 7.1.2 and let X : r0, T sˆΩ ÑHγ be the up to modifications unique pFtqtPr0,T s-predictable stochastic process whichsatisfies suptPr0,T s XtLppP;¨Hγ q
ă 8 and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.39)
Then
suptPr0,T s
›
›max
1, XtHγ
(›
›
LppP;|¨|qď?
2›
›max
1, ξHγ(›
›
LppP;|¨|q
¨ Ep1´ηq„
T 1´η?
2 F LGpHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q BLGpHγ ,HSpU,Hγ´η2qq
ă 8.(7.40)
Proof of Proposition 7.1.13. Observe that Theorem 6.2.32 implies that for all t P
7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 195
r0, T s it holds that
›
›max
1, XtHγ
(›
›
LppP;|¨|q
ď›
›max
1, X0Hγ
(›
›
LppP;|¨|q`
ż t
0
›
›eApt´sq F pXsq›
›
LppP;¨Hγ qds
`
c
p pp´ 1q
2
„ż t
0
›
›eApt´sqBpXsq›
›
2
LppP;¨HSpU,Hγ qqds
12
ď›
›max
1, X0Hγ
(›
›
LppP;|¨|q`
„
tp1´ηq
p1´ ηq
ż t
0
pt´ sq´η F pXsq2LppP;¨Hγ´η
qds
12
`
c
p pp´ 1q
2
„ż t
0
pt´ sq´η BpXsq2LppP;¨HSpU,Hγ´η2q
qds
12
.
(7.41)
This shows that for all t P r0, T s it holds that
›
›max
1, XtHγ
(›
›
LppP;|¨|q
ď›
›max
1, X0Hγ
(›
›
LppP;|¨|q`
„ż t
0
pt´ sq´η›
›max
1, XsHγ
(›
›
2
LppP;|¨|qds
12
¨
«
F LGpHγ ,Hγ´ηq
d
T p1´ηq
p1´ ηq` BLGpHγ ,HSpU,Hγ´η2qqq
c
p pp´ 1q
2
ff
.
(7.42)
This proves that for all t P r0, T s it holds that
›
›max
1, XtHγ
(›
›
2
LppP;|¨|qď 2
›
›max
1, X0Hγ
(›
›
2
LppP;|¨|q
`
ż t
0
pt´ sq´η›
›max
1, XsHγ
(›
›
2
LppP;|¨|qds
¨
«
F LGpHγ ,Hγ´ηq
d
2T p1´ηq
p1´ ηq` BLGpHγ ,HSpU,Hγ´η2qq
a
p pp´ 1q
ff2
.
(7.43)
An application of Corollary 1.4.6 hence completes the proof of Proposition 7.1.13.
196 CHAPTER 7. SOLUTIONS OF SPDES
Proposition 7.1.14 (Solution process of the SPDE enjoys more regularity than theintial value). Assume the setting in Subsection 7.1.2 and let X : r0, T s ˆ Ω Ñ Hγ bethe up to modifications unique pFtqtPr0,T s-predictable stochastic process which satisfiessuptPr0,T s XtLppP;¨Hγ q
ă 8 and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.44)
Then it holds for all t P r0, T s, r P rγ,mint1 ` γ ´ η, 12 ` βuq that P`
Xt ´ eAtX0 P
Hr
˘
“ 1 and
›
›Xt ´ eAtX0
›
›
LppP;¨Hr qď max
!
1, supsPr0,T s XsLppP;¨Hγ q
)
¨
«
F LippHγ ,Hγ´ηqtp1`γ´η´rq
p1` γ ´ η ´ rq`
a
p pp´ 1q BLippHγ ,HSpU,Hβqqtp12`β´rq
?2 p1` 2β ´ 2rq12
ff
ă 8
(7.45)
and it holds for all t P p0, T s, r P rγ,mint1 ` γ ´ η, 12 ` βuq that P`
Xt P Hr
˘
“ 1and
XtLppP;¨Hr qď
X0LppP;¨Hγ q
tpr´γq`max
!
1, supsPr0,T s XsLppP;¨Hγ q
)
¨
„
F LippHγ,Hγ´ηqtp1`γ´η´rq
p1`γ´η´rq`
?p pp´1q BLippHγ,HSpU,Hβqq
tp12`β´rq
?2 p1`2β´2rq12
ă 8.(7.46)
Proof of Proposition 7.1.14. First of all, recall that for all t P r0, T s it holds P-a.s.that
Xt ´ eAtX0 “
ż t
0
eApt´sqF pXsq ds`
ż t
0
eApt´sqBpXsq dWs. (7.47)
Moreover, note that Theorem 4.7.6 implies that for all t P r0, T s, r P rγ, γ ` 1´ ηq itholds that
ż t
0
›
›eApt´sqF pXsq›
›
LppP;¨Hr qds
ď F LippHγ ,Hγ´ηqmax
#
1, supsPr0,T s
XsLppP;¨Hγ q
+
„ż t
0
pt´ sqpγ´η´rq ds
“ F LippHγ ,Hγ´ηqmax
#
1, supsPr0,T s
XsLppP;¨Hγ q
+
tp1`γ´η´rq
p1` γ ´ η ´ rqă 8.
(7.48)
7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 197
Moreover, observe that Theorem 4.7.6 ensures that for all t P r0, T s, r P rγ, β ` 12q
it holds that„ż t
0
›
›eApt´sqBpXsq›
›
2
LppP;¨HSpU,Hrqqds
12
ď BLippHγ ,HSpU,Hβqqmax
#
1, supsPr0,T s
XsLppP;¨Hγ q
+
„ż t
0
pt´ sqp2β´2rq ds
12
ď BLippHγ ,HSpU,Hβqqmax
#
1, supsPr0,T s
XsLppP;¨Hγ q
+
tp12`β´rq
p1` 2β ´ 2rq12ă 8
(7.49)
Combining (7.47), (7.48), and (7.49) with Theorem 4.7.6 completes the proof ofProposition 7.1.14.
7.1.7 Temporal-regularity of solution processes of SPDEs
Exercise 7.1.15. Assume the setting in Subsection 7.1.2 and let X : r0, T s ˆ Ω Ñ
Hγ be the up to modifications unique pFtqtPr0,T s-predictable stochastic process whichsatisfies suptPr0,T s XtLppP;¨Hγ q
ă 8 and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.50)
Prove then that for all r P rγ,mint1`γ´η, 12`βuq, ε P`
0,mint1`γ´η´r, 12`β´ru˘
it holds that
supt1,t2Pr0,T st1‰t2
¨
˝
›
›
`
Xt1 ´ et1AX0
˘
´`
Xt2 ´ et2AX0
˘›
›
LppP;¨Hr q
|t1 ´ t2|ε
˛
‚ă 8. (7.51)
Exercise 7.1.16. Assume the setting in Subsection 7.1.2, let δ P rγ,8q, assumethat ξ P LppP; ¨Hδq, and let X : r0, T s ˆ Ω Ñ Hγ be the up to modifications uniquepFtqtPr0,T s-predictable stochastic process which satisfies suptPr0,T s XtLppP;¨Hγ q
ă 8
and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.52)
Prove then that for all r P rγ,mint1`γ´η, 12`βuq, ε P`
0,mint1`γ´η´r, 12`β´ru˘
it holds that
supt1,t2Pr0,T st1‰t2
¨
˝
|mintt1, t2u|maxtr`ε´δ,0u
Xt1 ´Xt2LppP;¨Hr q
|t1 ´ t2|ε
˛
‚ă 8. (7.53)
198 CHAPTER 7. SOLUTIONS OF SPDES
7.1.8 Existence of continuous solutions
See, e.g., Theorem 7.1 in Van Neerven et al. [28] for a more general result.
Proposition 7.1.17. Let pH, x¨, ¨yH , ¨Hq and pU, x¨, ¨yU , ¨Uq be separable R-Hilbertspaces, let H Ď H be a non-empty orthonormal basis of H, let T P p0,8q, ρ P R, letpΩ,F ,Pq be a probability space with a normal filtration pFtqtPr0,T s, let pWtqtPr0,T s be anIdU -cylindrical pFtqtPr0,T s-Wiener process, let A : DpAq Ď H Ñ H be a diagonal linearoperator which satisfies sup
`
t´1u Y σP pA ´ ρq˘
ă 0, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R,be a family of interpolation spaces associated to ρ ´ A, and let γ P R, η P r0, 1q,F P LippHγ, Hγ´ηq, B P LippHγ, HSpU,Hγ´η2qq, ξ P MpF0,BpHγqq. Then thereexists an pFtqtPr0,T s-adapted stochastic processes with continuous sample paths which
satisfies that for all t P r0, T s it holds P-a.s. that Xt “ etAξ `şt
0ept´sqAF pXsq ds `
şt
0ept´sqABpXsq dWs and which satisfies
suptPr0,T s
›
›max
1, XtHγ
(›
›
LppP;|¨|Rqď?
2›
›max
1, ξHγ(›
›
LppP;|¨|Rq
¨ Ep1´ηq„
T 1´η?
2 F LGpHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q BLGpHγ ,HSpU,Hγ´η2qq
.(7.54)
Proof of Proposition 7.1.17. Throughout this proof let Ωn P F0, n P N0, be thesets with the property that for all n P N0 it holds that Ωn “ tξHγ ă nu and letξn : Ω Ñ Hγ, n P N, be the mappings with the property that for all n P N it holdsthat ξn “ ξ1Ωn . Note that for all q P r0,8q, n P N it holds that E
“
ξnqHγ
‰
ď nq ă 8.Theorem 7.1.9, Exercise 7.1.15, and the Kolmogorov-Chentsov theorem hence ensurethat there exist pFtqtPr0,T s-adapted stochastic processes with continuous sample pathsXn : r0, T s ˆ Ω Ñ Hγ, n P N, which satisfy suptPr0,T sE
“
Xnt Hγ
‰
ă 8 and whichsatisfy that for all n P N, t P r0, T s it holds P-a.s. that
Xnt “ etAξn `
ż t
0
ept´sqAF pXns q ds`
ż t
0
ept´sqABpXns q dWs. (7.55)
Observe that for all k P N, n,m P tk, k ` 1, . . . u, t P r0, T s it holds P-a.s. that
1Ωk rXnt ´X
mt s “
ż t
0
ept´sqA1Ωk
“
F`
1ΩkXns
˘
´ F`
1ΩkXms
˘‰
ds
`
ż t
0
ept´sqA1Ωk
“
B`
1ΩkXns
˘
´B`
1ΩkXms
˘‰
dWs.
(7.56)
7.1. PROPERTIES OF MILD SOLUTIONS OF SPDES 199
We can hence apply Proposition 2.1 in [15] to obtain that for all k P N, n,m P
tk, k ` 1, . . . u it holds that
suptPr0,T s
1Ωk rXnt ´X
mt sLppP;Hγq
“ 0. (7.57)
This implies that
P
˜
@ k P N : @n,m P tk, k ` 1, . . . u : 1Ωk
«
suptPr0,T s
Xnt ´X
mt Hγ
ff
“ 0
¸
“ 1. (7.58)
Next let Y : r0, T s ˆ Ω Ñ Hγ be the mapping with the property that for all pt, ωq Pr0, T s ˆ Ω it holds that
Ytpωq “8ÿ
n“1
Xnt pωq ¨ 1ΩnzΩn´1pωq. (7.59)
Note that for all n P N it holds that
1Ωn suptPr0,T s
Yt ´Xnt Hγ “ sup
tPr0,T s
1ΩnYt ´ 1ΩnXnt Hγ
“ suptPr0,T s
›
›
›
›
›
«
nÿ
k“1
1ΩkzΩk´1Xkt
ff
´ 1ΩnXnt
›
›
›
›
›
Hγ
“ suptPr0,T s
›
›
›
›
›
nÿ
k“1
1ΩkzΩk´1
“
Xkt ´X
nt
‰
›
›
›
›
›
Hγ
“
nÿ
k“1
1ΩkzΩk´1
«
1Ωk suptPr0,T s
›
›Xkt ´X
nt
›
›
Hγ
ff
(7.60)
This and (7.58) show that
P
˜
@n P N : 1Ωn suptPr0,T s
Yt ´Xnt Hγ “ 0
¸
“ 1. (7.61)
Hence, we obtain that for all n P N, t P r0, T s it holds P-a.s. that
1ΩnYt “ 1ΩnXnt
“ 1Ωn
„
etAξn `
ż t
0
ept´sqAF pXns q ds`
ż t
0
ept´sqABpXns q dWs
“ 1Ωn
„
etAξ `
ż t
0
ept´sqA1ΩnF pXns q ds`
ż t
0
ept´sqA1ΩnBpXns q dWs
“ 1Ωn
„
etAξ `
ż t
0
ept´sqAF pYsq ds`
ż t
0
ept´sqABpYsq dWs
.
(7.62)
200 CHAPTER 7. SOLUTIONS OF SPDES
This implies that for all t P r0, T s it holds P-a.s. that
Yt “ etAξ `
ż t
0
ept´sqAF pYsq ds`
ż t
0
ept´sqABpYsq dWs. (7.63)
Next note that (7.61) and Proposition 7.1.13 ensure that for all n P N it holds that
suptPr0,T s
›
›max
1, 1ΩnYtHγ(›
›
LppP;|¨|q“ sup
tPr0,T s
›
›max
1, 1ΩnXnt Hγ
(›
›
LppP;|¨|q
ď suptPr0,T s
›
›max
1, Xnt Hγ
(›
›
LppP;|¨|qď?
2›
›max
1, ξnHγ(›
›
LppP;|¨|q
¨ Ep1´ηq„
T 1´η?
2 F LGpHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q BLGpHγ ,HSpU,Hγ´η2qq
.
(7.64)
This and Fatou’s lemma imply that for all t P r0, T s it holds that
›
›max
1, YtHγ(›
›
LppP;|¨|q“
›
›
›lim infnÑ8
max
1, 1ΩnYtHγ(
›
›
›
LppP;|¨|q
ď lim infnÑ8
›
›max
1, 1ΩnYtHγ(›
›
LppP;|¨|qď?
2›
›max
1, ξHγ(›
›
LppP;|¨|q
¨ Ep1´ηq„
T 1´η?
2 F LGpHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q BLGpHγ ,HSpU,Hγ´η2qq
.
(7.65)
The proof of Proposition 7.1.17 is thus completed.
7.2 Examples of SPDEs
7.2.1 Second order SPDEs
Let T, ϑ P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq“ pU, 〈¨, ¨〉U , ¨Uq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq
, ¨L2pBorelp0,1q;|¨|Rqq, ξ P
L2pP|F0 ; ¨Hq, let pWtqtPr0,T s be a cylindrical IdU -Wiener process w.r.t. pFtqtPr0,T s,let pekqkPN Ď H satisfy that for all k P N and Borelp0,1q-almost all x P p0, 1q it holdsthat
ekpxq “?
2 sinpkπxq, (7.66)
let A : DpAq Ď H Ñ H be a linear operator with the property that
DpAq “
#
v P H :8ÿ
k“1
k4|〈ek, v〉H |
2Ră 8
+
(7.67)
7.2. EXAMPLES OF SPDES 201
and with the property that for all v P DpAq it holds that
Av “8ÿ
k“1
´ϑπ2k2 〈ek, v〉H ek, (7.68)
let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to ´A,let f, b : p0, 1q ˆ RÑ R be Bpp0, 1q ˆRq/BpRq-measurable functions withż 1
0
|fpx, 0q|2 ` |bpx, 0q|2 dx` supxPp0,1q
supy1,y2PRy1‰y2
|fpx,y1q´fpx,y2q|`|bpx,y1q´bpx,y2q||y1´y2|
ă 8, (7.69)
and let F : H Ñ H be the function with the property that for all v P H andBorelp0,1q-almost all x P p0, 1q it holds that
`
F pvq˘
pxq “ fpx, vpxqq. Observe thatthe linear operator A is the Laplace operator with Dirichlet boundary conditions onL2pBorelp0,1q; |¨|Rq multiplied by ϑ. and let β P p´12,´14q. Next observe that for allv P H it holds that
8ÿ
k“1
bp¨, vp¨qq ekp¨q2Hβ“
8ÿ
k“1
›
›p´Aqβ`
bp¨, vp¨qq ekp¨q˘›
›
2
H
“
8ÿ
l,k“1
ˇ
ˇ
⟨el, p´Aq
β`
bp¨, vp¨qq ekp¨q˘⟩
H
ˇ
ˇ
2
R“
8ÿ
l,k“1
ˇ
ˇ
⟨p´Aqβel, bp¨, vp¨qq ekp¨q
⟩H
ˇ
ˇ
2
R
“
8ÿ
l,k“1
›
›p´Aqβel›
›
2
H|〈el, bp¨, vp¨qq ekp¨q〉H |
2R
“ÿ
lPN
›
›p´Aqβel›
›
2
H
«
ÿ
kPN
|〈ek, bp¨, vp¨qq elp¨q〉H |2
ff
“ÿ
lPN
›
›p´Aqβel›
›
2
Hbp¨, vp¨qq elp¨q
2H ď 2 bp¨, vp¨qq2H
«
ÿ
lPN
›
›p´Aqβel›
›
2
H
ff
“ 2›
›p´Aqβ›
›
2
HSpHqbp¨, vp¨qq2H ă 8.
(7.70)
This and Proposition 2.5.19 ensure that there exists a unique mapping B : H Ñ
HSpH,Hβq which satisfies that for all v P H, u P Cpr0, 1s,Rq and Borelp0,1q-almostall x P p0, 1q it holds that
`
Bpvqu˘
pxq “ bpx, vpxqq ¨ upxq. (7.71)
In addition, (7.70) implies that for all v P H it holds that
BpvqHSpU,Hβq ď?
2›
›p´Aqβ›
›
HSpHqbp¨, vp¨qqH ă 8. (7.72)
202 CHAPTER 7. SOLUTIONS OF SPDES
Moreover, note that for all v, w P H it holds that
Bpvq ´Bpwq2HSpU,Hβq “ÿ
kPN
rBpvq ´Bpwqs ek2Hβ
“ÿ
kPN
›
›p´Aqβ rrBpvq ´Bpwqs eks›
›
2
H
“ÿ
k,lPN
ˇ
ˇ
⟨el, p´Aq
βrrBpvq ´Bpwqs eks
⟩H
ˇ
ˇ
2
“ÿ
k,lPN
›
›p´Aqβel›
›
2
H|〈el, rBpvq ´Bpwqs ek〉H |
2 .
(7.73)
This proves that for all v, w P H it holds that
Bpvq ´Bpwq2HSpU,Hβq “ÿ
k,lPN
›
›p´Aqβel›
›
2
H|〈ek, rBpvq ´Bpwqs el〉H |
2
“ÿ
lPN
›
›p´Aqβel›
›
2
HrBpvq ´Bpwqs el
2H
ď 2ÿ
lPN
›
›p´Aqβel›
›
2
Hbp¨, vp¨qq ´ bp¨, wp¨qq2H
ď 2›
›p´Aqβ›
›
2
HSpHq
»
– supxPp0,1q
supy1,y2PRy1‰y2
|bpx, y1q ´ bpx, y2q|
|y1 ´ y2|
fi
fl
2
v ´ w2H ă 8.
(7.74)
This shows that the mapping B : H Ñ HSpH,Hβq in (7.71) is an element of the setC0,1pH,HSpH,Hβqq. We can hence apply Theorem 7.1.9 to obtain that there existsan up to modifications unique pFtqtPr0,T s-predictable stochastic process X : r0, T s ˆΩ Ñ H which satisfies suptPr0,T sE
“
Xt2H
‰
ă 8 and which is a mild solution of theSPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (7.75)
The stochastic process X is thus a mild solution of the SPDE
dXtpxq “”
ϑ B2
Bx2Xtpxq ` fpx,Xtpxqq
ı
dt` bpx,Xtpxqq dWtpxq, Xtp0q “ Xtp1q “ 0
(7.76)for x P p0, 1q, t P r0, T s and with X0pxq “ ξpxq for x P p0, 1q. For example, in thecase where b fulfills @x P p0, 1q, y P R : bpx, yq “ 1, the SPDE (7.76) reduces to theSPDE with additive noise
dXtpxq “”
ϑ B2
Bx2Xtpxq ` fpx,Xtpxqq
ı
dt` dWtpxq, Xtp0q “ Xtp1q “ 0 (7.77)
7.2. EXAMPLES OF SPDES 203
for x P p0, 1q, t P r0, T s and with X0pxq “ ξpxq for x P p0, 1q and in the case where fand b fulfill @x P p0, 1q, y P R : fpx, yq “ 0 and bpx, yq “ y, the SPDE (7.76) reducesto the stochastic heat equation with linear multiplicative noise
dXtpxq “”
ϑ B2
Bx2Xtpxq
ı
dt`Xtpxq dWtpxq (7.78)
for x P p0, 1q, t P r0, T s and with X0pxq “ ξpxq for x P p0, 1q. In the literature theSPDE (7.78) is referred to as continuous version of the parabolic Anderson model.
Question 7.2.1. Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, letpH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq
, ¨L2pBorelp0,1q;|¨|Rqq, let ξ P
H, let pWtqtPr0,T s be a cylindrical IdH-Wiener process w.r.t. pFtqtPr0,T s, let b : p0, 1q ˆRÑ R be a globally Lipschitz continuous function, let X : r0, T s ˆΩ Ñ H be a mildsolution process of the SPDE
dXtpxq “B2
Bx2Xtpxq dt` bpx,Xtpxqq dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ ξpxq
for x P p0, 1q, t P r0, T s, let A : DpAq Ď H Ñ H be the Laplace operator withDirichlet boundary conditions on H, and let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a familyof interpolation spaces associated to ´A.
(i) For which r P R does it holds that @ t P r0, T s : PpXt P Hrq “ 1?
(ii) For which r P R does it holds that @ t P p0, T s : PpXt P Hrq “ 1?
Exercise 7.2.2 (Variances). Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq
, ¨L2pBorelp0,1q;|¨|Rqq,
let pWtqtPr0,T s be a cylindrical IdH-Wiener process w.r.t. pFtqtPr0,T s, and let X : r0, T sˆΩ Ñ H be a mild solution process of the SPDE
dXtpxq “B2
Bx2Xtpxq dt` dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ 0 (7.79)
for x P p0, 1q, t P r0, T s. Prove or disprove the following statement: It holds thatş1
0E“
|XT pxq|2‰
dx “ř8
n“11´e´2π2n2T
π2n2 .
204 CHAPTER 7. SOLUTIONS OF SPDES
Part IV
Numerical Analysis of SPDEs
205
Chapter 8
Strong numerical approximationsfor SPDEs
Consider the setting of Section 7.1.2. If one wants to simulate a solution process ofan SPDE on a computer approximatively, then one needs to discretize the possiblyinfinite dimensional Hilbert space H (spatial approximations), the possibly infinitedimensional Hilbert space U (noise approximations) as well as the time intervalr0, T s (temporal approximations). Section 8.1 deals with spatial approximations forSPDEs, Section 8.2 analyses temporal numerical approximations for SPDEs, Sec-tion 8.3 considers noise approximations for SPDEs, and Section 8.4 combines spatial(Section 8.1), temporal (Section 8.2), and noise approximations (Section 8.3) to ob-tain full-discrete numerical approximations for SPDEs.
8.1 Spatial spectral Galerkin approximations for
SPDEs
8.1.1 Galerkin projections
We study Galerkin approximations in Hilbert spaces. For this we recall the conceptof a projection of a Hilbert space on a subspace in Definition 3.3.12 above.
207
208 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Lemma 8.1.1 (Representations of projections in Hilbert spaces). Let K P tR,Cu,let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, let U Ď H be a closed subspace of H, letB Ď U be an orthonormal basis of U , let PU P LpHq be the projection of H on U ,and let v P H. Then
PUpvq “ P spanpBq pvq “ÿ
bPB
〈b, v〉H b. (8.1)
8.1. SPATIAL SPECTRAL GALERKIN APPROXIMATIONS FOR SPDES 209
Example 8.1.2. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, letB Ď H be an orthonormal basis of H, let In Ď B, n P N, be a non-decreasingsequence of subsets of B which satisfies YnPNIn “ B, and let v P H. Then it holdsfor all n P N that
›
›
›
›
›
v ´
«
ÿ
bPIn
〈b, v〉H b
ff›
›
›
›
›
H
“
›
›
›
›
›
«
ÿ
bPB
〈b, v〉H b
ff
´
«
ÿ
bPIn
〈b, v〉H b
ff›
›
›
›
›
H
“
›
›
›
›
›
›
ÿ
bPBzIn
〈b, v〉H b
›
›
›
›
›
›
H
“
d
ÿ
bPBzIn
〈b, v〉H b2H “
d
ÿ
bPBzIn
|〈b, v〉H |2.
(8.2)
The fact thatř
bPB |〈b, v〉H |2“ v2H ă 8 and the assumption that In Ď B, n P N, is
non-decreasing with the property that YnPNIn “ B hence imply that
limnÑ8
›
›
›
›
›
v ´
«
ÿ
bPIn
〈b, v〉H b
ff›
›
›
›
›
H
“ 0. (8.3)
For sufficiently large n P N it thus holds thatÿ
bPIn
〈b, v〉H b (8.4)
is, in the sense of (8.3), a good approximation of v P H. For example, assume thatK “ R, assume that
pH, 〈¨, ¨〉H , ¨Hq “`
L2pBorelp0,1q; |¨|q, ¨L2pBorelp0,1q;|¨|q
, 〈¨, ¨〉L2pBorelp0,1q;|¨|q
˘
, (8.5)
let en P H, n P N, satisfy that for all n P N and Borelp0,1q-almost all x P p0, 1q it
holds that enpxq “?
2 sinpnπxq, assume that B “ te1, e2, . . . u and assume that forall n P N it holds that In “ te1, e2, . . . , enu. Then
limnÑ8
ż 1
0
ˇ
ˇ
ˇ
ˇ
vpxq ´nř
k“1
2 sinpkπxq
„
1
∫0
sinpkπyq vpyq dy
ˇ
ˇ
ˇ
ˇ
2
dx “ 0 (8.6)
and for sufficiently large n P N it holds that the function
nř
k“1
2 sinpkπxq
„
1
∫0
sinpkπyq vpyq dy
, x P p0, 1q, (8.7)
is, in the sense of (8.6), a good approximation of v P H.
210 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Example 8.1.2 above illustrates how suitable finite-dimensional approximationsof vectors in an infinite dimensional Hilbert space can be obtained; see (8.4) in Ex-ample 8.1.2. Equation (8.3) in Example 8.1.2 also shows that these approximationsin finite dimensional subspaces of the Hilbert space converge to the original vectorin the infinite dimensional Hilbert space. In Proposition 8.1.4 below we intend toprovide more information about how fast the finite dimensional approximations con-verge to the vector in the infinite dimensional Hilbert space. In the formulation ofProposition 8.1.4 the following lemma is used.
Lemma 8.1.3. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, letA : DpAq Ď H Ñ H be a symmetric diagonal linear operator with infpσP pAqq ą 0,let B Ď H be an orthonormal basis of H, let λ : B Ñ p0,8q be a function with theproperty that
DpAq “
#
v P H :ÿ
bPB
|λb 〈b, v〉H |2ă 8
+
(8.8)
and with the property that for all v P DpAq it holds that Av “ř
bPB λb 〈b, v〉H b, letpHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to A, andlet r P R, v P Hr, b P B. Then
〈b, v〉H b “@
bpλbqr
, vD
Hr
bpλbqr
P Hr. (8.9)
The proof of Lemma 8.1.3 is clear and therefore omitted. Instead we now formu-late the main result of this subsection.
8.1. SPATIAL SPECTRAL GALERKIN APPROXIMATIONS FOR SPDES 211
Proposition 8.1.4 (A central idea for spectral Galerkin approximations). Let K PtR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be an K-Hilbert space, let A : DpAq Ď H Ñ H be a sym-metric diagonal linear operator with infpσP pAqq ą 0, let B Ď H be an orthonormalbasis of H, let λ : BÑ p0,8q be a function with the property that
DpAq “
#
v P H :ÿ
bPB
|λb 〈b, v〉H |2ă 8
+
(8.10)
and with the property that for all v P DpAq it holds that Av “ř
bPB λb 〈b, v〉H b, letpHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of interpolation spaces associated to A, andlet r P R, ρ P r0,8q, I P PpBq, πI P LpHrq satisfy that for all v P Hr it holds that
πIpvq “ÿ
bPI
〈b, v〉H b “ÿ
bPI
@
bpλbqr
, vD
Hr
bpλbqr
“ PspanpIq
Hr pvq P Hr. (8.11)
Then it holds for all v P Hr`ρ that
v ´ πIpvqHr ď›
›A´ρ pIdHr ´πIq›
›
LpHrqvHr`ρ “
„
infbPBzI
λb
´ρ
vHr`ρ . (8.12)
Proof of Proposition 8.1.4. Observe that Proposition 3.5.5 implies that
›
›A´ρ pIdHr ´πIq›
›
LpHrq“
›
›
›
›
›
›
A´ρ
¨
˝
ÿ
bPBzI
@
bpλbqr
, p¨qD
Hr
bpλbqr
˛
‚
›
›
›
›
›
›
LpHrq
“
›
›
›
›
›
›
ÿ
bPBzI
1pλbqρ
@
bpλbqr
, p¨qD
Hr
bpλbqr
›
›
›
›
›
›
LpHrq
“ supbPBzI
”
1pλbqρ
ı
“
„
infbPBzI
λb
´ρ
.
(8.13)
The proof of Proposition 8.1.4 is thus completed.
In a number of cases the right hand side of estimate (8.12) converges to zero witha polynomial rate of convergence. This is illustrated in the next example.
212 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Example 8.1.5 (The Laplace operator with Dirichlet boundary conditions). LetA : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace operator withDirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R,be a family of interpolation spaces associated to ´A, let r P R, ρ P r0,8q, and leten P L
2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all n P N and Borelp0,1q-almost all
x P p0, 1q it holds that enpxq “?
2 sinpnπxq. Then Proposition 8.1.4 proves that forall v P Hr`ρ, n P N it holds that
›
›
›
›
›
v ´nÿ
k“1
〈ek, v〉H ek
›
›
›
›
›
Hr
ďvHr`ρπ2ρ n2ρ
ďvHr`ρn2ρ
. (8.14)
Note that, in the setting of Example 8.1.5, it holds for all v P Hr`ρ that
supnPN
˜
n2ρ
›
›
›
›
›
v ´nÿ
k“1
〈ek, v〉H ek
›
›
›
›
›
Hr
¸
ă 8. (8.15)
The polynomial convergence rate 2ρ in (8.14) and (8.15) can, in general, not beimproved. This is the subject of the next exercise.
Exercise 8.1.6 (Lower bounds on the convergence speed of spectral Galerkin projec-tions). Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace opera-tor with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let pHr, 〈¨, ¨〉Hr , ¨Hrq,r P R, be a family of interpolation spaces associated to ´A, let r P R, ρ P r0,8q, andlet en P L
2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all n P N and Borelp0,1q-almost all
x P p0, 1q it holds that enpxq “?
2 sinpnπxq. Give an example of a vector v P Hr`ρ
such that for all ε P p0,8q it holds that
supnPN
˜
np2ρ`εq
›
›
›
›
›
v ´nÿ
k“1
〈ek, v〉H ek
›
›
›
›
›
Hr
¸
“ 8. (8.16)
The next result, Corollary 8.1.7, specialises Proposition 8.1.4 to the case wherethe vector in the possibly infinite dimensional Hilbert space is the solution processof some SPDE at a fixed time instance.
8.1. SPATIAL SPECTRAL GALERKIN APPROXIMATIONS FOR SPDES 213
Corollary 8.1.7 (Galerkin projections for SPDEs). Assume the setting in Subsec-tion 7.1.2, let B Ď H be an orthonormal basis of H, let λ : BÑ p´8, 0q be a functionwith the property that
DpAq “
#
v P H :ÿ
bPB
|λb 〈b, v〉H |2ă 8
+
(8.17)
and with the property that for all v P DpAq it holds that Av “ř
bPB λb 〈b, v〉H b, letr P p´8, γs, I P PpBq, πI P LpHrq satisfy that for all v P Hr it holds that
πIpvq “ÿ
bPI
〈b, v〉H b “ÿ
bPI
@
b|λb|
r , vD
Hr
b|λb|
r “ PspanpIq
Hr pvq P Hr, (8.18)
and let X : r0, T s ˆ Ω Ñ Hγ be the up to modifications unique pFtqtPr0,T s-predictablestochastic process which satisfies suptPr0,T s XtLppP;¨Hγ q
ă 8 and which is a mildsolution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (8.19)
Then
suptPr0,T s
Xt ´ πIpXtqLppP;¨Hr qď
„
infbPBzI
|λb|
pr´γq
XCpr0,T s,LppP;¨Hγ qqă 8. (8.20)
Corollary 8.1.7 is an immediate consequence from Proposition 8.1.4 and Propo-sition 7.1.11.
Question 8.1.8. Let f : p0, 1q Ñ R be the function with the property that for allx P p0, 1q it holds that fpxq “ x. For which r P R does it hold that
supNPN
˜
N r
ż 1
0
ˇ
ˇ
ˇ
ˇ
fpxq ´Nř
n“1
2 sinpnπxq1ş
0
sinpnπyq fpyq dy
ˇ
ˇ
ˇ
ˇ
2
dx
¸
ă 8? (8.21)
Question 8.1.9. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq and letv P DpAq. For which r P R does it hold that
supNPN
˜
N r
ż 1
0
ˇ
ˇ
ˇ
ˇ
vpxq ´Nř
n“1
2 sinpnπxq1ş
0
sinpnπyq vpyq dy
ˇ
ˇ
ˇ
ˇ
2
dx
¸
ă 8? (8.22)
214 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
8.1.2 Setting
Assume the setting in Section 7.1.2, let B Ď H be an orthonormal basis of H, letλ : BÑ R be a function, assume that
DpAq “
#
v P H :ÿ
bPB
|λb 〈eb, v〉H |2ă 8
+
, (8.23)
assume that for all v P DpAq it holds that Av “ř
bPB λb 〈b, v〉H b, and let pπIqIPPpBq ĎLpHγ´ηq satisfy that for all v P Hγ´η, I P PpBq it holds that
πIpvq “ÿ
bPI
〈b, v〉H b. (8.24)
The above setting allows us to apply Theorem 7.1.9 to obtain that there existup to modifications unique pFtqtPr0,T s-predictable stochastic processes XI : r0, T s ˆΩ Ñ πIpHγq, I P PpBq, with the property that for all I P PpBq it holds thatsuptPr0,T s X
It LppP;¨Hγ q
ă 8 and with the property that for all I P PpBq, t P r0, T sit holds P-a.s. that
XIt “ eAtπIpξq `
ż t
0
eApt´sqπI`
F pXIs q˘
ds`
ż t
0
eApt´sqπI`
BpXIs q dWs
˘
. (8.25)
8.1.3 A strong numerical approximation result for spectralGalerkin approximations of SPDEs
Lemma 8.1.10. Assume the setting in Section 8.1.2 and let I, J P PpBq. Then
›
›XI´XJ
›
›
Cpr0,T s,LppP;¨Hγ qqď?
2›
›πIzJXI` πJzIX
J›
›
Cpr0,T s,LppP;¨Hγ qq(8.26)
¨ Ep1´ηq„
T 1´η?
2 |πIXJF |C0,1pHγ,Hγ´ηq?1´η
`a
ppp´ 1qT 1´η |πIXJB|C0,1pHγ ,HSpU,Hγ´η2qq
ă 8.
Proof of Lemma 8.1.10. Let F P CpHγ, Hγ´ηq and B P CpHγ, HSpU,Hγ´η2qq be thefunctions with the property that for all v P Hγ, u P U it holds that
F pvq “ πIXJ`
F pvq˘
and Bpvqu “ πIXJ`
Bpvqu˘
. (8.27)
8.1. SPATIAL SPECTRAL GALERKIN APPROXIMATIONS FOR SPDES 215
The identity πIπJ “ πIXJ then shows that for all t P r0, T s it holds P-a.s. that
„
XIt ´
ż t
0
eApt´sqF pXIs q ds´
ż t
0
eApt´sqBpXIs q dWs
`
„ż t
0
eApt´sqF pXJs q ds`
ż t
0
eApt´sqBpXJs q dWs ´X
Jt
“
„
XIt ´ πJ
ˆ
eAtπIpξq `
ż t
0
eApt´sqπIF pXIs q ds`
ż t
0
eApt´sqπIBpXIs q dWs
˙
`
„
πI
ˆ
eAtπJpξq `
ż t
0
eApt´sqπJF pXJs q ds`
ż t
0
eApt´sqπJBpXJs q dWs
˙
´XJt
““
IdHγ ´πJ‰
XIt `
“
πI ´ IdHγ‰
XJt “ πIzJX
It ´ πJzIX
Jt .
(8.28)
An application of Proposition 7.1.4 hence proves that
suptPr0,T s
›
›XIt ´X
Jt
›
›
LppP;¨Hγ q
ď Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
ppp´ 1qT 1´η |B|C0,1pHγ ,HSpU,Hγ´η2qq
¨?
2 suptPr0,T s
›
›πIzJXIt ´ πJzIX
Jt
›
›
LppP;¨Hγ q.
(8.29)
This and the identity
suptPr0,T s
›
›πIzJXIt ´ πJzIX
Jt
›
›
LppP;¨Hγ q“ sup
tPr0,T s
›
›πIzJXIt ` πJzIX
Jt
›
›
LppP;¨Hγ q(8.30)
complete the proof of Lemma 8.1.10.
The next result, Corollary 8.1.11, is an immediate consequence of Lemma 8.1.10above.
Corollary 8.1.11. Assume the setting in Section 8.1.2, let r P pγ,mintγ`1´η, β`12uq, I P PpBq, and assume that ξ P LppP; ¨Hrq. Then
suptPr0,T s
›
›XB
t ´XIt
›
›
LppP;¨Hγ qď?
2“
infbPBzI |λb|‰pγ´rq
suptPr0,T s
›
›XB
t
›
›
LppP;¨Hr q(8.31)
¨ Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
ppp´ 1qT 1´η |B|C0,1pHγ ,HSpU,Hγ´η2qq
ă 8.
216 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
The next result, Corollary 8.1.12, establishes a certain uniform moment boundfor the processes XI , I P PpBq. Corollary 8.1.12 is an immediate consequence ofProposition 7.1.11 and from Theorem 7.1.9.
Corollary 8.1.12. Assume the setting in Section 8.1.2, let r P rγ,mintγ`1´η, β`12uq, ρ P
“
maxtr ` η ´ γ, 2pr ´ βqu, 1˘
, and assume that ξ P LppP; ¨Hrq. Then
supIPPpBq
›
›XI›
›
Cpr0,T s,LppP;¨Hr qqď?
2 max
1, ξLppP;¨Hr q
(
¨ Ep1´ρq„
T 1´ρ?
2 F LippHr,Hr´ρq?1´ρ
`a
T 1´ρppp´ 1q BLippHr,HSpU,Hr´ρ2qq
ă 8.(8.32)
Lemma 8.1.13. Assume the setting in Section 8.1.2, let r P rγ,mintγ`1´η, β`12uq,I, J P PpBq, and assume that ξ P LppP; ¨Hrq. Then
›
›XI´XJ
›
›
Cpr0,T s,LppP;¨Hγ qqď 2
›
›p´Aqpγ´rqπI4J›
›
LpHqmaxKPtI,Ju
›
›XK›
›
Cpr0,T s,LppP;¨Hr qq
¨ Ep1´ηq„
T 1´η?
2 |πIXJF |C0,1pHγ,Hγ´ηq?1´η
`a
ppp´ 1qT 1´η |πIXJB|C0,1pHγ ,HSpU,Hγ´η2qq
ă 8.
Proof of Lemma 8.1.13. Observe that
›
›πIzJXI` πJzIX
J›
›
2
Cpr0,T s,LppP;¨Hγ qq“ sup
tPr0,T s
›
›πIzJXIt ` πJzIX
Jt
›
›
2
LppP;¨Hγ q
“ suptPr0,T s
›
›p´Aqpγ´rqπI4J p´Aqpr´γq
“
πIzJXIt ` πJzIX
Jt
‰›
›
2
LppP;¨Hγ q
ď›
›p´Aqpγ´rqπI4J›
›
2
LpHqsuptPr0,T s
›
›πIzJXIt ` πJzIX
Jt
›
›
2
LppP;¨Hr q
“›
›p´Aqpγ´rqπI4J›
›
2
LpHqsuptPr0,T s
›
›
›
›
›πIzJXIt ` πJzIX
Jt
›
›
2
Hr
›
›
›
Lp2pP;|¨|q
“›
›p´Aqpγ´rqπI4J›
›
2
LpHqsuptPr0,T s
›
›
›
›
›πIzJXIt
›
›
2
Hr`›
›πJzIXJt
›
›
2
Hr
›
›
›
Lp2pP;|¨|q
ď›
›p´Aqpγ´rqπI4J›
›
2
LpHqsuptPr0,T s
„
›
›
›
›
›πIzJXIt
›
›
2
Hr
›
›
›
Lp2pP;|¨|q`
›
›
›
›
›πJzIXJt
›
›
2
Hr
›
›
›
Lp2pP;|¨|q
“›
›p´Aqpγ´rqπI4J›
›
2
LpHqsuptPr0,T s
”
›
›πIzJXIt
›
›
2
LppP;¨Hr q`›
›πJzIXJt
›
›
2
LppP;¨Hr q
ı
.
(8.33)
Combining this with Lemma 8.1.10 completes the proof of Lemma 8.1.13.
8.1. SPATIAL SPECTRAL GALERKIN APPROXIMATIONS FOR SPDES 217
Let us illustrate a bit different perspective on Lemma 8.1.13. For this assume thesetting in Section 8.1.2, let dr : PpBq ˆ PpBq Ñ r0,8q, r P p0,8q, be the mappingswith the property that for all I, J P PpBq, r P p0,8q it holds that
drpI, Jq “›
›p´Aq´rπI4J›
›
LpHq, (8.34)
let r P pγ,mint1`γ´η, β` 12uq, and let ξ P LppP; ¨Hrq. Exercise 8.1.14 then showsthat the pair pPpBq, drq is a metric space and Lemma 8.1.13, in particular, ensuresthat the mapping1
pPpBq, dr´γq Q I ÞÑ XIP
´
Cpr0, T s, LppP; ¨Hγ qq, ¨Cpr0,T s,LppP;¨Hγ qq
¯
(8.35)
is globally Lipschitz continuous with a Lipschitz constant which is smaller or equalthan
2 ¨
«
supIPPpBq
suptPr0,T s
›
›XI›
›
LppP;¨Hr q
ff
(8.36)
¨ Ep1´ηq„
T 1´η?
2 |πIXJF |CC1pHγ,Hγ´ηq?1´η
`a
ppp´ 1qT 1´η |πIXJB|C0,1pHγ ,HSpU,Hγ´η2qq
ă 8.
Exercise 8.1.14. Assume the setting in Section 8.1.2 and let dr : PpBq ˆ PpBq Ñr0,8q, r P p0,8q, be the mappings with the property that for all I, J P PpBq, r Pp0,8q it holds that
drpI, Jq “›
›p´Aq´rπI4J›
›
LpHq. (8.37)
Prove that for every r P p0,8q it holds that the pair pPpBq, drq is a metric space.
1Clearly, the domain (the set of arguments) of the mapping (8.35) is not pPpBq, dr´γq but theset PpBq. The notation (8.35) is nonetheless used in order to emphasize the specific metric definedon the set PpBq of arguments. The same comment applies to the co-domain of the mapping (8.35).
218 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Exercise 8.1.15. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq (see Def-inition 3.5.7), let T “ 1, let en P L2pBorelp0,1q; |¨|Rq, n P N, satisfy that for all
n P N and Borelp0,1q-almost all x P p0, 1q it holds that enpxq “?
2 sinpnπxq, letpΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s be a cylindrical IdH-Wienerprocess w.r.t. pFtqtPr0,T s, let X : r0, T sˆΩ Ñ H be an pFtqtPr0,T s-predictable stochasticprocess which fulfills that for all t P r0, T s it holds P-a.s. that
Xt “
ż t
0
eApt´sq dWs, (8.38)
let πN P LpHq, N P N, satisfy that for all N P N, v P H it holds that πNpvq “řNn“1 〈en, v〉H en, and let XN : r0, T s ˆΩ Ñ PNpHq, N P N, be pFtqtPr0,T s-predictable
stochastic processes which satisfy that for all N P N, t P r0, T s it holds P-a.s. that
XNt “
ż t
0
πN eApt´sq dWs. (8.39)
Write a Matlab function which plots Monte Carlo approximations of the real num-
bers`
E“
X223
T ´ XNT
2H
‰˘12for N P t20, 21, 22, 23, . . . , 218, 219u. Hint: Use the fact
that for every N P N you can simulate exactly from XNT pPq.
Question 8.1.16 (Convergence speed of spectral Galerkin approximations). Assumethe setting in Section 8.1.2 and assume that ξ P LppP; ¨Hγ`1
q. For which r P R does
it holds that there exists a real number C P R such that for all I P PpBq it holds that
suptPr0,T s
›
›XB
t ´XIt
›
›
LppP;¨Hγ qď C
“
infbPBzI |λb|‰´r
. (8.40)
8.2 Temporal numerical approximations for SPDEs
In this section we present and analyze a few temporal numerical approximations forSPDEs. For this the following notation is used.
Definition 8.2.1 (Round down to the grid). Let t¨uh : R Ñ R, h P p0,8q, be themappings with the property that for all h P p0,8q, t P R it holds that
ttuh “ max`
p´8, ts X t0, h,´h, 2h,´2h, . . . u˘
. (8.41)
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 219
Definition 8.2.2 (Down to the grid). Let z¨h : RÑ R, h P p0,8q, be the mappingswith the property that for all h P p0,8q, t P R it holds that
zth “ max`
p´8, tq X t0, h,´h, 2h,´2h, . . . u˘
. (8.42)
Using Definition 8.2.1 we will now present in Subsections 8.2.1 and 8.2.3 be-low a few temporal numerical approximation methods for SPDEs and then analyzethe strong approximation errors of one of these approximation methods in Subsec-tion 8.2.4 below.
8.2.1 Euler type approximations for SPDEs
8.2.1.1 Exponential Euler method
Definition 8.2.3 (Exponential Euler approximations). Assume the setting in Sec-tion 7.1.2, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “ ξ and which fulfills that for all n Pt0, 1, . . . , N ´ 1u it holds P-a.s. that
Yn`1 “ eATN
˜
Yn ` F`
Yn˘
TN`
ż pn`1qTN
nTN
B`
Yn˘
dWs
¸
. (8.43)
Then we call Y an exponential Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.44)
with time step size TN.
Definition 8.2.4 (Naturally-interpolated exponential Euler approximations). As-sume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ be anpFtqtPr0,T s-adapted stochastic process which fulfills Y0 “ ξ and which fulfills that forall t P p0, T s it holds P-a.s. that
Yt “ eApt´zthq
ˆ
Yzth ` F`
Yzth
˘
pt´ zthq `
ż t
zth
B`
Yzth
˘
dWs
˙
. (8.45)
Then we call Y a naturally-interpolated exponential Euler approximation for theSPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.46)
with time step size h.
220 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ
be a naturally-interpolated exponential Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.47)
with time step size h. Then note that for all t P r0, T s it holds P-a.s. that
Yt “ eAtξ `
ż t
0
eApt´tsuhqF`
Ytsuh
˘
ds`
ż t
0
eApt´tsuhqB`
Ytsuh
˘
dWs. (8.48)
Exercise 8.2.5. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let T P
p0,8q, N P N, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s be a cylin-drical IdH-Wiener process w.r.t. pFtqtPr0,T s, let Y : r0, T s ˆ Ω Ñ H be a naturally-interpolated exponential Euler approximation for the SPDE
dXt “ AXt dt` dWt, t P r0, T s, X0 “ 0 (8.49)
with time step size TN, and let X : r0, T s ˆ Ω Ñ H be an pFtqtPr0,T s-predictablestochastic process which fulfills that for all t P r0, T s it holds P-a.s. that
Xt “
ż t
0
eApt´sq dWs. (8.50)
Prove that for all r P r0, 14q it holds that`
E“
XT ´ YT 2H
‰˘12ď T r
p1´4rqNr .
8.2.1.2 Accelerated exponential Euler method
Definition 8.2.6 (Accelerated exponential Euler approximations). Assume the set-ting in Section 7.1.2, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be anpFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “ ξ and which fulfillsthat for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that
Yn`1 “ eATN Yn ` A
´1´
eATN ´ IdH
¯
F`
Yn˘
`
ż pn`1qTN
nTN
eAppn`1qTN´sqB
`
Yn˘
dWs.
(8.51)
Then we call Y an accelerated exponential Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.52)
with time step size TN.
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 221
Definition 8.2.7 (Naturally-interpolated accelerated exponential Euler approxima-tions). Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ
be an pFtqtPr0,T s-adapted stochastic process which fulfills Y0 “ ξ and which fulfills thatfor all t P p0, T s it holds P-a.s. that
Yt “ eApt´zthq Yzth `
ż t
zth
eApt´sqF`
Yzth
˘
ds`
ż t
zth
eApt´sqB`
Yzth
˘
dWs. (8.53)
Then we call Y a naturally-interpolated accelerated exponential Euler approximationfor the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.54)
with time step size h.
Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ
be a naturally-interpolated exponential Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.55)
with time step size h. Then note that for all t P r0, T s it holds P-a.s. that
Yt “ eAtξ `
ż t
0
eApt´sqF`
Ytsuh
˘
ds`
ż t
0
eApt´sqB`
Ytsuh
˘
dWs. (8.56)
8.2.1.3 Linear-implicit Euler method
Definition 8.2.8 (Linear-implicit Euler approximations). Assume the setting in Sec-tion 7.1.2, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic processes which fulfills Y0 “ ξ and which fulfills that for alln P t0, 1, . . . , N ´ 1u it holds P-a.s. that
Yn`1 “`
IdH ´TNA˘´1
˜
Yn ` F`
Yn˘
TN`
ż pn`1qTN
nTN
B`
Yn˘
dWs
¸
. (8.57)
Then we call Y a linear-implicit Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.58)
with time step size TN.
222 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Definition 8.2.9 (Naturally-interpolated linear-implicit Euler approximations). As-sume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ be anpFtqtPr0,T s-adapted stochastic processes which fulfills Y0 “ ξ and which fulfills that forall t P p0, T s it holds P-a.s. that
Yt “ pIdH ´pt´ zthqAq´1
ˆ
Yzth ` F`
Yzth
˘
pt´ zthq `
ż t
zth
B`
Yzth
˘
dWs
˙
. (8.59)
Then we call Y a naturally-interpolated linear-implicit Euler approximation for theSPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.60)
with time step size h.
Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ
be a naturally-interpolated linear-implicit Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.61)
with time step size h. Then note that for all t P r0, T s it holds P-a.s. that
Yt “`
IdH ´pt´ ttuhqA˘´1 `
IdH ´hA˘´ttuhh
ξ
`
ż t
0
`
IdH ´pt´ ttuhqA˘´1 `
IdH ´hA˘ptsuh´ttuhqh
F`
Ytsuh
˘
ds
`
ż t
0
`
IdH ´pt´ ttuhqA˘´1 `
IdH ´hA˘ptsuh´ttuhqh
B`
Ytsuh
˘
dWs.
(8.62)
8.2.1.4 Linear-implicit Crank-Nicolson-Euler method
Definition 8.2.10 (Linear-implicit Crank-Nicolson-Euler approximations). Assumethe setting in Section 7.1.2, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be anpFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “ ξ and which fulfillsthat for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that
Yn`1 “`
IdH ´T
2NA˘´1
˜
`
IdH `T
2NA˘
Yn ` F pYnqTN`
ż pn`1qTN
nTN
BpYnq dWs
¸
.
(8.63)
Then we call Y a linear-implicit Crank-Nicolson-Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.64)
with time step size TN.
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 223
Definition 8.2.11 (Naturally-interpolated linear-implicit Crank-Nicolson-Euler ap-proximations). Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ be an pFtqtPr0,T s-adapted stochastic processes which fulfills Y0 “ ξ and whichfulfills that for all t P p0, T s it holds P-a.s. that
Yt “´
IdH ´pt´zthq
2A¯´1
ˆ
´
IdH `pt´zthq
2A¯
Yzth ` F`
Yzth
˘
pt´ zthq `
ż t
zth
B`
Yzth
˘
dWs
˙
.
(8.65)
Then we call Y a naturally-interpolated linear-implicit Crank-Nicolson-Euler approx-imation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.66)
with time step size h.
Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ
be a naturally-interpolated linear-implicit Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.67)
with time step size h. Then note that for all t P r0, T s it holds P-a.s. that
Yt “`
IdH ´pt´ttuhq
2A˘´1 `
IdH ´h2A˘´ttuhh
ξ
`
ż t
0
`
IdH ´pt´ttuhq
2A˘´1 `
IdH ´h2A˘ptsuh´ttuhqh
“
12AYtsuh ` F
`
Ytsuh
˘‰
ds
`
ż t
0
`
IdH ´pt´ttuhq
2A˘´1 `
IdH ´h2A˘ptsuh´ttuhqh
B`
Ytsuh
˘
dWs.
(8.68)
8.2.2 Nonlinearity-stopped Euler type approximations forSPDEs
The next result is a special case of Theorem 2.1 in Hutzenthaler et al. [12] (see also[13]).
224 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Theorem 8.2.12 (Strong and weak divergence of the Euler method for SDEs withsuperlinearly growing coefficients). Let T, ε, p P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq bea stochastic basis, let W : r0, T s ˆ Ω Ñ R be a standard Brownian motion w.r.t.pFtqtPr0,T s, let µ, σ PMpBpRq,BpRqq, ξ PMpF0,BpRqq satisfy P
`
σpξq ‰ 0˘
ą 0, letY N : t0, 1, . . . , Nu ˆΩ Ñ R, N P N, satisfy that for all N P N, n P t0, 1, . . . , N ´ 1uit holds that Y N
0 “ ξ and
Y Nn`1 “ Y N
n ` µpY Nn q
TN` σpY N
n q`
Wpn`1qTN ´WnTN
˘
, (8.69)
and assume that for all x P p´8, 1εsYr1ε,8q it holds that |µpxq|`|σpxq| ě ε |x|p1`εq.Then limNÑ8 E
“
|Y NN |
p‰
“ 8.
8.2.2.1 Nonlinearity-stopped exponential Euler method
Definition 8.2.13 (Nonlinearity-stopped exponential Euler approximations). As-sume the setting in Section 7.1.2, let N P N, α P rγ´η, γs, assume that F pHγq Ď Hα,and let Y : t0, 1, . . . , NuˆΩ Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic processwhich fulfills Y0 “ ξ and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s.that
Yn`1 “ 1tF pYnq2HαąNTuYn
` 1tF pYnq2HαďNTueA
TN
˜
Yn ` F`
Yn˘
TN`
ż pn`1qTN
nTN
B`
Yn˘
dWs
¸
.(8.70)
Then we call Y a nonlinearity-stopped exponential Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.71)
with time step size TN.
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 225
Definition 8.2.14 (Naturally-interpolated nonlinearity-stopped exponential Eulerapproximations). Assume the setting in Section 7.1.2, let h P p0,8q, α P rγ ´ η, γs,assume that F pHγq Ď Hα, and let Y : r0, T s ˆ Ω Ñ Hγ be an pFtqtPr0,T s-adaptedstochastic process which fulfills Y0 “ ξ and which fulfills that for all t P p0, T s it holdsP-a.s. that
Yt “ 1tF pYzthq2Hα
ą1hu Yzth (8.72)
` 1tF pYzthq2αď1hu e
Apt´zthq
ˆ
Yzth ` F`
Yzth
˘
pt´ zthq `
ż t
zth
B`
Yzth
˘
dWs
˙
.
Then we call Y a naturally-interpolated nonlinearity-stopped exponential Euler ap-proximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.73)
with time step size h.
8.2.2.2 Nonlinearity-stopped linear-implicit Euler method
Definition 8.2.15 (Nonlinearity-stopped linear-implicit Euler approximations). As-sume the setting in Section 7.1.2, let N P N, α P rγ´η, γs, assume that F pHγq Ď Hα,and let Y : t0, 1, . . . , NuˆΩ Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic processwhich fulfills Y0 “ ξ and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s.that
Yn`1 “ 1tF pYnq2HαąNTuYn (8.74)
` 1tF pYnq2HαďNTu
`
IdH ´TNA˘´1
˜
Yn ` F`
Yn˘
TN`
ż pn`1qTN
nTN
B`
Yn˘
dWs
¸
.
Then we call Y a nonlinearity-stopped linear-implicit Euler approximation for theSPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.75)
with time step size TN.
226 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Definition 8.2.16 (Naturally-interpolated nonlinearity-stopped linear-implicit Eu-ler approximations). Assume the setting in Section 7.1.2, let h P p0,8q, α P rγ´η, γs,assume that F pHγq Ď Hα, and let Y : r0, T s ˆ Ω Ñ Hγ be an pFtqtPr0,T s-adaptedstochastic processes which fulfills Y0 “ ξ and which fulfills that for all t P p0, T s itholds P-a.s. that
Yt “ 1tF pYzthq2Hα
ą1hu Yzth ` 1tF pYzthq2Hα
ď1hu pIdH ´pt´ zthqAq´1
ˆ
Yzth
` F`
Yzth
˘
pt´ zthq `
ż t
zth
B`
Yzth
˘
dWs
˙
. (8.76)
Then we call Y a naturally-interpolated nonlinearity-stopped linear-implicit Eulerapproximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.77)
with time step size h.
8.2.3 Milstein type approximations for SPDEs
8.2.3.1 Exponential Milstein method
Definition 8.2.17 (Exponential Milstein approximations). Assume the setting inSection 7.1.2, assume that γ “ β, assume that B : Hγ Ñ HSpU,Hγq is continu-ously Frechet differentiable, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be anpFnTNqnPt0,1,...,Nu-adapted stochastic processes which fulfills Y0 “ ξ and which fulfillsthat for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that
Yn`1 “ eATN
˜
Yn ` F`
Yn˘
TN`
żpn`1qTN
nTN
B`
Yn˘
dWs
`
żpn`1qTN
nTN
B1`
Yn˘
ˆż s
nTN
B`
Yn˘
dWu
˙
dWs
¸
.
(8.78)
Then we call Y an exponential Milstein approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.79)
with time step size TN.
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 227
Definition 8.2.18 (Naturally-interpolated exponential Milstein approximations).Assume the setting in Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ
HSpU,Hγq is continuously Frechet differentiable, let h P p0,8q, and let Y : r0, T s ˆΩ Ñ Hγ be an pFtqtPr0,T s-adapted stochastic process which fulfills Y0 “ ξ and whichfulfills that for all t P p0, T s it holds P-a.s. that
Yt “ eApt´zthq
ˆ
Yzth ` F`
Yzth
˘
pt´ zthq
`
ż t
zth
”
B`
Yzth
˘
`B1`
Yzth
˘s
∫zth
B`
Yzth
˘
dWu
ı
dWs
˙
.
(8.80)
Then we call Y a naturally-interpolated exponential Milstein approximation for theSPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.81)
with time step size h.
Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ
be a naturally-interpolated exponential Milstein approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.82)
with time step size h. Then note that for all t P r0, T s it holds P-a.s. that
Yt “ eAtξ `
ż t
0
eApt´tsuhqF`
Ytsuh
˘
ds
`
ż t
0
eApt´tsuhq”
B`
Ytsuh
˘
`B1`
Ytsuh
˘s
∫tsuh
BpYtsuhq dWu
ı
dWs.
(8.83)
228 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
8.2.3.2 Linear-implicit Milstein method
Definition 8.2.19 (Linear-implicit Milstein approximations). Assume the settingin Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ HSpU,Hγq is contin-uously Frechet differentiable, let N P N, and let Y : t0, 1, . . . , Nu ˆ Ω Ñ Hγ be anpFnTNqnPt0,1,...,Nu-adapted stochastic processes which fulfills Y0 “ ξ and which fulfillsthat for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that
Yn`1 “`
IdH ´TNA˘´1
˜
Yn ` F`
Yn˘
TN`
żpn`1qTN
nTN
B`
Yn˘
dWs
`
żpn`1qTN
nTN
B1`
Yn˘
ˆż s
nTN
B`
Yn˘
dWu
˙
dWs
¸
.
(8.84)
Then we call Y a linear-implicit Milstein approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.85)
with time step size TN.
Definition 8.2.20 (Naturally-interpolated linear-implicit Milstein approximations).Assume the setting in Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ
HSpU,Hγq is continuously Frechet differentiable, let h P p0,8q, and let Y : r0, T s ˆΩ Ñ Hγ be an pFtqtPr0,T s-adapted stochastic processes which fulfills Y0 “ ξ and whichfulfills that for all t P p0, T s it holds P-a.s. that
Yt “pIdH ´pt´ zthqAq´1
ˆ
Yzth ` F`
Yzth
˘
pt´ zthq
`
ż t
zth
”
B`
Yzth
˘
`B1`
Yzth
˘s
∫zth
B`
Yzth
˘
dWu
ı
dWs
˙
.
(8.86)
Then we call Y a naturally-interpolated linear-implicit Milstein approximation forthe SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.87)
with time step size h.
Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T s ˆ Ω Ñ Hγ
be a naturally-interpolated linear-implicit Milstein approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.88)
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 229
with time step size h. Then note that for all t P r0, T s it holds P-a.s. that
Yt “`
IdH ´pt´ ttuhqA˘´1 `
IdH ´hA˘´ttuhh
ξ
`
ż t
0
`
IdH ´pt´ ttuhqA˘´1 `
IdH ´hA˘ptsuh´ttuhqh
F`
Ytsuh
˘
ds
`
ż t
0
`
IdH ´pt´ ttuhqA˘´1 `
IdH ´hA˘ptsuh´ttuhqh
B`
Ytsuh
˘
dWs
`
ż t
0
`
IdH ´pt´ ttuhqA˘´1 `
IdH ´hA˘ptsuh´ttuhqh
”
B1`
Yttuh
˘s
∫ttuh
B`
Yttuh
˘
dWu
ı
dWs.
(8.89)
8.2.3.3 Linear-implicit Crank-Nicolson-Milstein method
Definition 8.2.21 (Linear-implicit Crank-Nicolson-Milstein approximations). As-sume the setting in Section 7.1.2, assume that γ “ β, assume that B : Hγ Ñ
HSpU,Hγq is continuously Frechet differentiable, let N P N, and let Y : t0, 1, . . . , NuˆΩ Ñ Hγ be an pFnTNqnPt0,1,...,Nu-adapted stochastic processes which fulfills Y0 “ ξ andwhich fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that
Yn`1 “`
IdH ´T
2NA˘´1
˜
`
IdH `T
2NA˘
Yn ` F`
Yn˘
TN`
żpn`1qTN
nTN
B`
Yn˘
dWs
`
żpn`1qTN
nTN
B1`
Yn˘
ˆż s
nTN
B`
Yn˘
dWu
˙
dWs
¸
. (8.90)
Then we call Y a naturally-interpolated linear-implicit Crank-Nicolson-Milstein ap-proximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.91)
with time step size h.
230 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Definition 8.2.22 (Naturally-interpolated linear-implicit Crank-Nicolson-Milsteinapproximations). Assume the setting in Section 7.1.2, assume that γ “ β, assumethat B : Hγ Ñ HSpU,Hγq is continuously Frechet differentiable, let h P p0,8q, andlet Y : r0, T s ˆ Ω Ñ Hγ be an pFtqtPr0,T s-adapted stochastic processes which fulfillsY0 “ ξ and which fulfills that for all t P p0, T s it holds P-a.s. that
Yt “`
IdH ´12pt´ zthqA
˘´1
ˆ
Yzth `12AYzth pt´ zthq ` F
`
Yzth
˘
pt´ zthq
`
ż t
zth
”
B`
Yzth
˘
`B1`
Yzth
˘s
∫zth
B`
Yzth
˘
dWu
ı
dWs
˙
.
(8.92)
Then we call Y a naturally-interpolated linear-implicit Crank-Nicolson-Milstein ap-proximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.93)
with time step size h.
Assume the setting in Section 7.1.2, let h P p0,8q, and let Y : r0, T sˆΩ Ñ Hγ bea naturally-interpolated linear-implicit Crank-Nicolson-Milstein approximation forthe SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.94)
with time step size h. Then note that for all t P r0, T s it holds P-a.s. that
Yt “`
IdH ´pt´ttuhq
2A˘´1 `
IdH ´h2A˘´ttuhh
ξ
`
ż t
0
`
IdH ´pt´ttuhq
2A˘´1 `
IdH ´h2A˘ptsuh´ttuhqh
”
12AYtsuh ` F
`
Ytsuh
˘
ı
ds
`
ż t
0
`
IdH ´pt´ttuhq
2A˘´1 `
IdH ´h2A˘ptsuh´ttuhqh
B`
Ytsuh
˘
dWs
`
ż t
0
`
IdH ´pt´ttuhq
2A˘´1 `
IdH ´h2A˘ptsuh´ttuhqh
”
B1`
Yttuh
˘s
∫ttuh
B`
Yttuh
˘
dWu
ı
dWs.
(8.95)
8.2.4 Strong convergence analysis for exponential Euler ap-proximations
In this subsection we establish strong convergence with suitable rates of convergenceof exponential Euler approximations; see Definition 8.2.3. In this subsection wemainly follow the analysis in Kurniawan [21].
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 231
Lemma 8.2.23 (Regularity for the numerical approximations). Assume the settingin Subsection 7.1.2, let N P N, and let Y : r0, T sˆΩ Ñ Hγ be a naturally-interpolatedexponential Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.96)
with time step size TN. Then
suptPr0,T s
YtLppP;¨Hγ qď max
1,?
2 ξLppP;¨Hγ q
(
(8.97)
¨ E1´η
„
T 1´η?
2 F LippHγ,Hγ´ηq?1´η
`a
T 1´η p pp´ 1q BLippHγ ,HSpU,Hγ´η2qq
ă 8.
Proof of Lemma 8.2.23. Theorem 4.7.6 and Holder’s inequality imply that for allt P r0, T s it holds that
YtLppP;¨Hγ qď ξLppP;¨Hγ q
` F LippHγ ,Hγ´ηq
„
tp1´ηq
p1´ηq
ż t
0
pt´ sq´η max
1,›
›YtsuT N
›
›
2
LppP;¨Hγ q
(
ds
12
` BLippHγ ,HSpU,Hγ´η2qq
„
ppp´1q2
ż t
0
pt´ sq´η max
1,›
›YtsuT N
›
›
2
LppP;¨Hγ q
(
ds
12
.
(8.98)
This proves that for all t P r0, T s it holds that
YtLppP;¨Hγ qď ξLppP;¨Hγ q
`
„ż t
0
pt´ sq´η max
1,›
›YtsuT N
›
›
2
LppP;¨Hγ q
(
ds
12
¨
„
F LippHγ ,Hγ´ηqTp1´ηq2?
1´η` BLippHγ ,HSpU,Hγ´η2qq
?ppp´1q?
2
.
(8.99)
Induction hence proves that for all t P r0, T s it holds that YtLppP;¨Hγ qă 8.
Moreover, combining (8.99) with the estimate that for all a, b P R it holds thatpa` bq2 ď 2a2 ` 2b2 shows that for all t P r0, T s it holds that
max
1, Yt2LppP;¨Hγ q
(
ď max
1, 2 ξ2LppP;¨Hγ q
(
`
ż t
0
pt´ sq´η max
1,›
›YtsuT N
›
›
2
LppP;¨Hγ q
(
ds
¨
”
F LippHγ ,Hγ´ηq
?2T
p1´ηq2?
1´η` BLippHγ ,HSpU,Hγ´η2qq
a
p pp´ 1qı2
.
(8.100)
232 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Next note that for all u P r0, T s it holds that
suptPr0,us
ż t
0
pt´ sq´η max
1,›
›YtsuT N
›
›
2
LppP;¨Hγ q
(
ds
ď suptPr0,us
ż t
0
pt´ sq´η”
supvPr0,ss max
1, Yv2LppP;¨Hγ q
(
ı
ds
“ suptPr0,us
ż u
u´t
pt´ rs´ pu´ tqsq´η”
supvPr0,s´pu´tqs max
1, Yv2LppP;¨Hγ q
(
ı
ds
“ suptPr0,us
ż u
u´t
pu´ sq´η”
supvPr0,s`t´us max
1, Yv2LppP;¨Hγ q
(
ı
ds
ď suptPr0,us
ż u
u´t
pu´ sq´η”
supvPr0,ss max
1, Yv2LppP;¨Hγ q
(
ı
ds
“
ż u
0
pu´ sq´η”
supvPr0,ss max
1, Yv2LppP;¨Hγ q
(
ı
ds.
(8.101)
Putting this into (8.100) proves that for all u P r0, T s it holds that
suptPr0,us max
1, Yt2LppP;¨Hγ q
(
ď max
1, 2 ξ2LppP;¨Hγ q
(
`
ż u
0
pu´ sq´η”
suptPr0,ss max
1, Yt2LppP;¨Hγ q
(
ı
ds
¨
”
F LippHγ ,Hγ´ηq
?2T
p1´ηq2?
1´η` BLippHγ ,HSpU,Hγ´η2qq
a
p pp´ 1qı2
.
(8.102)
Combining this with Corollary 1.4.6 completes the proof of Lemma 8.2.23.
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 233
Lemma 8.2.24 (More regularity for the numerical approximations). Assume thesetting in Subsection 7.1.2, let N P N, and let Y : r0, T s ˆ Ω Ñ Hγ be a naturally-interpolated exponential Euler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.103)
with time step size TN. Then it holds for all t P r0, T s, r P rγ,mint1`γ´η, 12`βuqthat P
`
Yt ´ eAtY0 P Hr
˘
“ 1 and
›
›Yt ´ eAtY0
›
›
LppP;¨Hr qď max
#
1, supnPt0,1,...,Nu
YnTNLppP;¨Hγ q
+
¨
«
F LippHγ ,Hγ´ηqtp1`γ´η´rq
p1` γ ´ η ´ rq`
a
p pp´ 1q BLippHγ ,HSpU,Hβqqtp12`β´rq
p2` 4β ´ 4rq12
ff
ă 8
(8.104)
and it holds for all t P p0, T s, r P rγ,mint1` γ´ η, 12`βuq that P`
Yt P Hr
˘
“ 1 and
YtLppP;¨Hr qď
X0LppP;¨Hγ q
tpr´γq`max
!
1, supnPt0,1,...,Nu YnTNLppP;¨Hγ q
)
¨
„
F LippHγ,Hγ´ηqtp1`γ´η´rq
p1`γ´η´rq`
?p pp´1q BLippHγ,HSpU,Hβqq
tp12`β´rq
p2`4β´4rq12
ă 8.(8.105)
Proof of Lemma 8.2.24. First of all, recall that for all t P r0, T s it holds P-a.s. that
Yt ´ eAtY0 “
ż t
0
eApt´tsuT N qF pYtsuT Nq ds`
ż t
0
eApt´tsuT N qBpYtsuT Nq dWs. (8.106)
Moreover, note that Theorem 4.7.6 implies that for all t P r0, T s, r P rγ, γ ` 1´ ηq itholds that
ż t
0
›
›
›eApt´tsuT N qF pYtsuT N
q
›
›
›
LppP;¨Hr qds
ď F LippHγ ,Hγ´ηqmax
#
1, supsPr0,T s
YtsuT NLppP;¨Hγ q
+
„ż t
0
pt´ sqpγ´η´rq ds
“ F LippHγ ,Hγ´ηqmax
#
1, supsPr0,T s
YtsuT NLppP;¨Hγ q
+
tp1`γ´η´rq
p1` γ ´ η ´ rqă 8.
(8.107)
234 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Furthermore, observe that Theorem 4.7.6 ensures that for all t P r0, T s, r P rγ, β`12q
it holds that
„ż t
0
›
›
›eApt´tsuT N qBpYtsuT N
q
›
›
›
2
LppP;¨HSpU,Hrqqds
12
ď BLippHγ ,HSpU,Hβqqmax
#
1, supsPr0,T s
YtsuT NLppP;¨Hγ q
+
„ż t
0
pt´ sqp2β´2rq ds
12
ď BLippHγ ,HSpU,Hβqqmax
#
1, supsPr0,T s
YtsuT NLppP;¨Hγ q
+
tp12`β´rq
p1` 2β ´ 2rq12ă 8.
(8.108)
Combining (8.106), (8.107) and (8.108) with Theorem 4.7.6 and Theorem 6.2.32completes the proof of Lemma 8.2.24.
Lemma 8.2.25 (Regularity of a time integral associated to the numerical approx-imations). Assume the setting in Section 7.1.2, let ρ P r0, 1q, q P r1,8q, and forevery N P N let Y N : r0, T s ˆ Ω Ñ Hγ be a naturally-interpolated exponential Eulerapproximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.109)
with time step size TN. Then it holds for all r P p´8,mintp1´ρqq, 1´ η, 12` β ´ γuqthat
supNPN
suptPr0,T s
„
N rt
∫0pt´ sq´ρ
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
q
LppP;¨Hγ qds
1q
ă 8. (8.110)
Proof of Lemma 8.2.25. First of all, observe that for all N P N it holds that
suptPr0,TNs
„
t
∫0pt´ sq´ρ
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
q
LppP;¨Hγ qds
1q
ď
2 suptPp0,T s›
›Y Nt
›
›
LppP;¨Hγ qT p1´ρqq
p1´ ρq1qN p1´ρqq.
(8.111)
Next note that for all r P rγ,mint1 ` γ ´ η, 12 ` βuq, N P N, t P rTN, T s it holds
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 235
that
„
t
∫0pt´ sq´ρ
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
q
LppP;¨Hγ qds
1q
ď 2 supsPp0,T s
›
›Y Ns
›
›
LppP;¨Hγ q
„
TN
∫0pt´ sq´ρ ds
1q
`
«
supsPrTN,T s
`
tsuT N˘pr´γq
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
LppP;¨Hγ q
ff«
t
∫TN
pt´ sq´ρ
`
tsuT N˘q pr´γq
ds
ff1q
ď 2 supsPp0,T s
›
›Y Ns
›
›
LppP;¨Hγ q
T p1´ρqq
N p1´ρqq p1´ ρq1q
`
«
supsPrTN,T s
`
tsuT N˘pr´γq
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
LppP;¨Hγ q
ff
„
t´TN
∫0
pt´ TN ´ sq´ρ
sq pr´γqds
1q
.
(8.112)
This implies that that for all r P rγ,mint1` γ´ η, 12`βuq, N P N, t P rTN, T s withρ` qpr ´ γq ď 1 it holds that
„
t
∫0pt´ sq´ρ
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
q
LppP;¨Hγ qds
1q
ď
2 supsPp0,T s›
›Y Ns
›
›
LppP;¨Hγ qT p1´ρqq
N p1´ρqq p1´ ρq1q
`
«
supsPrTN,T s
`
tsuT N˘pr´γq
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
LppP;¨Hγ q
ff
¨ rt´ TNsr1´ρq´pr´γqs
“
B`
1´ ρ, 1´ qpr ´ γq˘‰1q
.
(8.113)
Combining this with (8.111) proves that for all r P rγ,mintγ`p1´ρqq, γ`1´η, 12`βuq,
236 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
N P N it holds that
suptPr0,T s
„
t
∫0pt´ sq´ρ
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
q
LppP;¨Hγ qds
1q
ď
2 supsPp0,T s›
›Y Ns
›
›
LppP;¨Hγ qT p1´ρqq
N p1´ρqq p1´ ρq1q
`
«
supsPrTN,T s
`
tsuT N˘pr´γq
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
LppP;¨Hγ q
ff
¨ T rγ`p1´ρqq´rs
“
B`
1´ ρ, 1´ qpr ´ γq˘‰1q
.
(8.114)
In the next step we observe that Theorem 6.2.32, Lemma 4.7.7 and Lemma 8.2.24ensure that for all N P N, s P rTN, T s, r P rγ,mint1` γ ´ η, 12` βuq it holds that
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
LppP;¨Hγ qď
›
›
›
`
eAps´tsuT N q ´ IdH˘
Y NtsuT N
›
›
›
LppP;¨Hγ q
`
ż s
tsuT N
›
›eAps´tuuT N qF pY NtuuT N
q›
›
LppP;¨Hγ qdu
`
«
p pp´ 1q
2
ż s
tsuT N
›
›
›eAps´tuuT N qBpY N
tuuT Nq
›
›
›
2
LppP;¨HSpU,Hγ qqdu
ff12
ď`
s´ tsuT N˘pr´γq
›
›
›Y N
tsuT N
›
›
›
LppP;¨Hr q
`
ż s
tsuT N
ps´ uq´η›
›
›F pY N
tuuT Nq
›
›
›
LppP;¨Hγ´ηqdu
`
«
p pp´ 1q
2
ż s
tsuT N
ps´ uqp2β´2γq›
›
›BpY N
tuuT Nq
›
›
›
2
LppP;¨HSpU,Hβqqdu
ff12
ă 8.
(8.115)
This implies that for all N P N, s P rTN, T s, r P rγ,mint1` γ ´ η, 12` βuq it holdsthat›
›
›Y Ns ´ Y N
tsuT N
›
›
›
LppP;¨Hγ qď
„
T
N
pr´γqˇ
ˇtsuT Nˇ
ˇ
pγ´rq
«
supuPp0,T s
upr´γq›
›Y Nu
›
›
LppP;¨Hr q
ff
`“
TN
‰p1´ηq F LippHγ,Hγ´ηq
p1´ηqmax
"
1, supnPt1,2,...,Nu›
›Y NnTN
›
›
LppP;¨Hγ q
*
`“
TN
‰p12`β´γq BLippHγ,HSpU,Hβqq
p1`2β´2γq12
”
p pp´1q2
ı12
max
"
1, supnPt1,2,...,Nu›
›Y NnTN
›
›
LppP;¨Hγ q
*
.
(8.116)
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 237
This shows that for all N P N, r P rγ,mint1` γ ´ η, 12` βuq it holds that
supsPrTN,T s
„
ˇ
ˇtsuT Nˇ
ˇ
pr´γq›
›
›Y Ns ´ Y N
tsuT N
›
›
›
LppP;¨Hγ q
ď
”
“
TN
‰pr´γq`“
TN
‰p1´ηq Tpr´γq F LippHγ,Hγ´ηq
p1´ηq
`“
TN
‰p12`β´γq Tpr´γq BLippHγ,HSpU,Hβqq
p12`β´γq12
”
p pp´1q4
ı12 ı
¨max
#
1, supvPrγ,rs
supuPp0,T s
upv´γq›
›Y Nu
›
›
LppP;¨Hv q
+
.
(8.117)
Putting this into (8.114) proves that for all N P N, r P rγ,mintγ ` p1´ρqq, γ ` 1 ´η, 12` βuq it holds that
suptPr0,T s
„
t
∫0pt´ sq´ρ
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
q
LppP;¨Hγ qds
1q
ď
«
2T p1´ρqq
N p1´ρqq p1´ ρq1q
`
”
1`T p1´ηq F LippHγ,Hγ´ηq
p1´ηq`
T p12`β´γq BLippHγ,HSpU,Hβqq
p12`β´γq12
”
p pp´1q4
ı12 ı
¨T p1´ρqq
N pr´γq
“
B`
1´ ρ, 1´ qpr ´ γq˘‰1q
ff
¨max
#
1, supvPrγ,rs
supuPp0,T s
upv´γq›
›Y Nu
›
›
LppP;¨Hv q
+
.
(8.118)
Hence, we obtain that for all N P N, r P rγ,mintγ ` p1´ρqq, γ ` 1 ´ η, 12 ` βuq itholds that
suptPr0,T s
„
t
∫0pt´ sq´ρ
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
q
LppP;¨Hγ qds
1q
ď
„
52` F LippHγ ,Hγ´ηq ` BLippHγ ,HSpU,Hβqq
?p pp´1q
2
¨
“
B`
1´ ρ, 1´ qpr ´ γq˘‰1q
maxpT 2, 1q
mint1´ ρ, 1´ η, 12` β ´ γuN pr´γqmax
#
1, supvPrγ,rs
supuPp0,T s
upv´γq›
›Y Nu
›
›
LppP;¨Hv q
+
.
(8.119)
This and Lemma 8.2.24 complete the proof of Lemma 8.2.25.
In the next result, Corollary 8.2.26, an estimate for the strong approximationerror of exponential Euler approximations is presented.
238 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Corollary 8.2.26. Assume the setting in Section 7.1.2, let X : r0, T s ˆ Ω Ñ Hγ bethe up to modifications unique pFtqtPr0,T s-predictable stochastic process which satisfiessuptPr0,T s XtLppP;¨Hγ q
ă 8 and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ, (8.120)
let N P N, and let Y : r0, T s ˆ Ω Ñ Hγ be a naturally-interpolated exponential Eulerapproximation for the SPDE (8.120) with time step size TN. Then
suptPr0,T s
Xt ´ YtLppP;¨Hγ q
ď Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qq
¨?
2 suptPr0,T s
«
ż t
0
›
›
›eApt´sq
”
F pYsq ´ eAps´tsuT N qF pYtsuT N
q
ı›
›
›
LppP;¨Hγ qds
`
„
p pp´1q2
t
∫0
›
›
›eApt´sq
”
BpYsq ´ eAps´tsuT N qBpYtsuT N
q
ı›
›
›
2
LppP;¨HSpU,Hγ qqds
12ff
ă 8.
(8.121)
Corollary 8.2.26 is an immediate consequence of Proposition 7.1.4 and of Theo-rem 6.2.32.
Theorem 8.2.27. Assume the setting in Section 7.1.2, let X : r0, T s ˆ Ω Ñ Hγ bethe up to modifications unique pFtqtPr0,T s-predictable stochastic process which satisfiessuptPr0,T s XtLppP;¨Hγ q
ă 8 and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ, (8.122)
and for every N P N let Y N : r0, T sˆΩ Ñ Hγ be a naturally-interpolated exponentialEuler approximation for the SPDE (8.122) with time step size TN. Then it holds forall r P
`
´8,mint1´ η, 12` β ´ γu˘
that
supNPN
suptPr0,T s
„
N r›
›Xt ´ YNt
›
›
LppP;¨Hγ q
ă 8. (8.123)
8.2. TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES 239
Proof of Theorem 8.2.27. Observe that Theorem 4.7.6 and Lemma 4.7.7 imply thatfor all t P r0, T s, ε P p0, 1´ ηq it holds that
ż t
0
›
›
›eApt´sq
”
F pY Ns q ´ e
Aps´tsuT N qF pY NtsuT N
q
ı›
›
›
LppP;¨Hγ qds
ď
ż t
0
›
›
›eApt´sq
”
F pY Ns q ´ F pY
NtsuT N
q
ı›
›
›
LppP;¨Hγ qds
`
ż t
0
›
›
›eApt´sq
`
IdH ´eAps´tsuT N q
˘
F pY NtsuT N
q
›
›
›
LppP;¨Hγ qds
ď
ż t
0
pt´ sq´η›
›
›F pY N
s q ´ F pYN
tsuT Nq
›
›
›
LppP;¨Hγ´η qds
`
ż t
0
pt´ sq´η´ε`
s´ tsuT N˘ε›
›
›F pY N
tsuT Nq
›
›
›
LppP;¨Hγ´η qds.
(8.124)
This ensures that for all t P r0, T s, ε P p0, 1´ ηq it holds that
ż t
0
›
›
›eApt´sq
”
F pY Ns q ´ e
Aps´tsuT N qF pY NtsuT N
q
ı›
›
›
LppP;¨Hγ qds
ď |F |C0,1pHγ ,Hγ´ηq
ż t
0
pt´ sq´η›
›
›Y Ns ´ Y N
tsuT N
›
›
›
LppP;¨Hγ qds
`
„
T
N
ε«
supsPr0,T s
›
›
›F pY N
tsuT Nq
›
›
›
LppP;¨Hγ´η q
ff
T p1´η´εq
p1´ η ´ εq.
(8.125)
Furthermore, Theorem 4.7.6 and Lemma 4.7.7 prove that for all t P r0, T s, ε P
240 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
p0, 12` β ´ γq it holds that
„
t
∫0
›
›
›eApt´sq
”
BpY Ns q ´ e
Aps´tsuT N qBpY NtsuT N
q
ı›
›
›
2
LppP;¨HSpU,Hγ qqds
12
ď
«
t
∫0pt´ sqp2β´2γq
›
›
›BpY N
s q ´BpYN
tsuT Nq
›
›
›
2
LppP;¨HSpU,Hβqqds
ff12
`
«
t
∫0pt´ sqp2β´2γ´2εq
`
s´ tsuT N˘2ε
›
›
›BpY N
tsuT Nq
›
›
›
2
LppP;¨HSpU,Hβqqds
ff12
ď |B|C0,1pHγ ,HSpU,Hβqq
„
t
∫0pt´ sqp2β´2γq
›
›
›Y Ns ´ Y N
tsuT N
›
›
›
2
LppP;¨Hγ qds
12
`
„
T
N
ε«
supsPr0,T s
›
›
›BpY N
tsuT Nq
›
›
›
LppP;¨HSpU,Hβqq
ff
T p12`β´γ´εq
p1` 2β ´ 2γ ´ 2εq12.
(8.126)
Combining (8.125), (8.126), Corollary 8.2.26, and Lemma 8.2.25 completes the proofof Theorem 8.2.27.
Question 8.2.28 (Convergence speed of exponential Euler approximations). LetT P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq “pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq
, ¨L2pBorelp0,1q;|¨|Rqq, let pWtqtPr0,T s be a cylindri-
cal IdH-Wiener process w.r.t. pFtqtPr0,T s, let b : p0, 1q ˆRÑ R be a globally Lipschitzcontinuous function, let ξ P H, let X : r0, T s ˆ Ω Ñ H be a mild solution process ofthe SPDE
dXtpxq “B2
Bx2Xtpxq dt` bpx,Xtpxqq dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ ξpxq
(8.127)for x P p0, 1q, t P r0, T s, assume that suptPr0,T sE
“
Xt2H
‰
ă 8, and for every N P N
let Y N : r0, T s ˆΩ Ñ H be a naturally-interpolated exponential Euler approximationfor the SPDE (8.127) with time step size TN.
(i) For which r P R does it holds that there exist a real number C P R such thatfor all N P N it holds that E
“
XT ´ YNT H
‰
ď CN´r?
(ii) For which r P R does it holds that for every p P p0,8q there exist a real numberC P R such that for all N P N it holds that suptPr0,T s
›
›Xt ´ YNt
›
›
LppP;¨Hqď
CN´r?
8.3. NOISE APPROXIMATIONS FOR SPDES 241
8.3 Noise approximations for SPDEs
8.3.1 Noise perturbation estimates
The next result, Corollary 8.3.1, is an immediate consequence of Proposition 7.1.4,Theorem 4.7.6, and Theorem 6.2.32.
Corollary 8.3.1 (Noise perturbation). Assume the setting in Section 7.1.2, let θ Prγ ´ β, 12q, B P C0,1pHγ, HSpU,Hβqq, and let X, X : r0, T s ˆ Ω Ñ Hγ be stochasticprocesses which satisfy supsPr0,T s
“
XsLppP;¨Hγ q` XsLppP;¨Hγ q
‰
ă 8, which satisfy
that X is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ, (8.128)
and which satisfy that X is a mild solution of the SPDE
dXt ““
AXt ` F pXtq‰
dt` BpXtq dWt, t P r0, T s, X0 “ ξ. (8.129)
Then
suptPr0,T s
›
›Xt ´ Xt
›
›
LppP;¨Hγ q
ď Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qq
¨T p12´θq
a
p pp´ 1q?
1´ 2θ
«
suptPp0,T q
›
›BpXsq ´ BpXsq›
›
LppP;¨HSpU,Hγ´θqq
ff
ă 8.
(8.130)
Proof. Proposition 7.1.4 ensures that
suptPr0,T s
›
›X1t ´X
2t
›
›
LppP;¨Hγ q
ď Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qq
¨?
2 suptPr0,T s
›
›
›
›
„
Xt ´
ż t
0
eApt´sqF pXsq ds´
ż t
0
eApt´sqBpXsq dWs
`
„ż t
0
eApt´sqF pXsq ds`
ż t
0
eApt´sqBpXsq dWs ´ Xt
›
›
›
›
LppP;¨Hγ q
ă 8.
(8.131)
242 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
This implies that
suptPr0,T s
›
›X1t ´X
2t
›
›
LppP;¨Hγ q
ď Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qq
¨?
2 suptPr0,T s
›
›
›
›
ż t
0
eApt´sq“
BpXsq ´ BpXsq‰
dWs
›
›
›
›
LppP;¨Hγ q
ă 8.
(8.132)
Theorem 6.2.32. and Theorem 4.7.6 hence prove that
suptPr0,T s
›
›X1t ´X
2t
›
›
LppP;¨Hγ q
ď Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qq
¨?
2
«
p pp´1q2
suptPr0,T s
ż t
0
s´2θds
ff12 «
supsPp0,T q
›
›BpXsq ´ BpXsq›
›
LppP;¨HSpU,Hγ´θqq
ff
ă 8.
(8.133)
This completes the proof of Corollary 8.3.1.
8.3.2 Noise approximations for SPDEs
The next result, Corollary 8.3.2, is an immediate consequence from Corollary 8.3.1above.
8.3. NOISE APPROXIMATIONS FOR SPDES 243
Corollary 8.3.2 (Noise discretizations). Assume the setting in Section 7.1.2, letθ P rγ ´ β, 12q, R, R P LpUq, and let X, X : r0, T s ˆ Ω Ñ Hγ be stochastic processeswhich satisfy supsPr0,T s
“
XsLppP;¨Hγ q` XsLppP;¨Hγ q
‰
ă 8, which satisfy that X is
a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtqRdWt, t P r0, T s, X0 “ ξ, (8.134)
and which satisfy that X is a mild solution of the SPDE
dXt ““
AXt ` F pXtq‰
dt`BpXtq R dWt, t P r0, T s, X0 “ ξ. (8.135)
Then
suptPr0,T s
›
›Xt ´ Xt
›
›
LppP;¨Hγ qď
«
supvPHγ
BpvqrR ´ RsHSpU,Hγ´θq
maxt1, vHγu
ff
¨ Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qqRLpUq
¨T p12´θq
a
p pp´ 1q?
1´ 2θ
«
suptPr0,T s
›
›maxt1, XsHγu›
›
LppP;|¨|q
ff
ă 8.
(8.136)
The next result, Corollary 8.3.3, illustrates how strong convergence rates for noisediscretizations can be obtained.
244 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
Corollary 8.3.3. Assume the setting in Section 7.1.2, let θ P rγ ´ β, 12q, r P
r0,8q, pRNqNPN0 Ď LpUq satisfy supNPN supvPHγNrBpvqrR0´RN sHSpU,Hγ´θq
maxt1,vHγ uă 8, and
let XN : r0, T s ˆ Ω Ñ Hγ, N P N0, be stochastic processes with the property that@N P N0 : supsPr0,T s XsLppP;¨Hγ q
ă 8 and with the property that for all N P N0 it
holds that XN is a mild solution of the SPDE
dXNt “
“
AXNt ` F pX
Nt q
‰
dt`BpXNt qRN dWt, t P r0, T s, X0 “ ξ. (8.137)
Then
suptPr0,T s
›
›X0t ´X
Nt
›
›
LppP;¨Hγ qď
1
N r
«
supMPN
supvPHγ
M rBpvqrR0 ´RM sHSpU,Hγ´θq
maxt1, vHγu
ff
¨ Ep1´ηq„
T 1´η?
2 |F |C0,1pHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q |B|C0,1pHγ ,HSpU,Hγ´η2qqR0LpUq
¨T p12´θq
a
p pp´ 1q?
1´ 2θ
«
suptPr0,T s
supMPN
›
›maxt1, XMs Hγu
›
›
LppP;|¨|q
ff
ă 8.
(8.138)
Proof of Corollary 8.3.3. First, observe that Proposition 7.1.13 ensures that for allM P N0 it holds that
suptPr0,T s
›
›max
1, XMt Hγ
(›
›
LppP;|¨|qď?
2›
›max
1, ξHγ(›
›
LppP;|¨|q
¨ Ep1´ηq„
T 1´η?
2 F LGpHγ,Hγ´ηq?1´η
`a
T 1´ηppp´ 1q Bp¨qRMLGpHγ ,HSpU,Hγ´η2qq
ă 8.
(8.139)
Next note that the assumption that supNPN supvPHγNrBpvqrR0´RN sHSpU,Hγ´θq
maxt1,vHγ uă 8
implies that
supMPN0
Bp¨qRMLGpHγ ,HSpU,Hγ´η2qq“ sup
MPN0
supvPHγ
«
BpvqRMHSpU,Hγ´η2q
max
1, vHγ(
ff
ď supvPHγ
«
BpvqHSpU,Hγ´η2qR0LpUq
max
1, vHγ(
ff
` supMPN0
supvPHγ
«
BpvqrR0 ´RM sHSpU,Hγ´η2q
max
1, vHγ(
ff
ă 8.
(8.140)
8.4. FULL DISCRETIZATIONS FOR SPDES 245
This and (8.139) prove that
supMPN0
suptPr0,T s
›
›max
1, XMt Hγ
(›
›
LppP;|¨|qă 8. (8.141)
This and Corollary 8.3.2 complete the proof of Corollary 8.3.3.
8.4 Full discretizations for SPDEs
8.4.1 Setting
Assume the setting in Section 7.1.2, let B Ď H be an orthonormal basis of H, letU Ď U be an orthonormal basis of U , let λ : B Ñ R be a function, assume thatDpAq “
v P H :ř
bPB |λb 〈b, v〉H |2ă 8
(
, assume that for all v P DpAq it holds thatAv “
ř
bPB λb 〈b, v〉H b and let pπIqIPPpBq Ď LpHγ´ηq and p$IqIPPpUq Ď LpUq satisfythat for all v P Hγ´η, u P U , I P PpBq, J P PpUq it holds that
πIpvq “ÿ
bPI
〈b, v〉H b and $Jpuq “ÿ
bPJ
〈b, u〉U b. (8.142)
8.4.2 Full-discrete spectral Galerkin exponential Euler methodfor SPDEs
Definition 8.4.1 (Full discrete spectral Galerkin exponential Euler approximations).Assume the setting in Section 8.4.1, let N P N, I P PpBq, J P PpUq and letY : t0, 1, . . . , Nu ˆ Ω Ñ πIpHγq be an pFnTNqnPt0,1,...,Nu-adapted stochastic processwhich fulfills Y0 “ πIpξq and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holdsP-a.s. that
Yn`1 “ eATN
˜
Yn ` πIpF pYnqqTN`
ż pn`1qTN
nTN
πI`
BpYnq$JpdWsq˘
¸
. (8.143)
Then we call Y a full-discrete spectral Galerkin exponential Euler approximation forthe SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.144)
with time step size TN, spatial approximation I and noise approximation J .
246 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
1 function Y = ExpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;6 y = y + f ( y )∗T/N + b( y ) . ∗dW;7 Y = exp( A∗T/N ) .∗ i d s t ( y ) / sqrt ( 2 ) ;8 end9 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;
10 end
Matlab code 8.1: A Matlab function for simulating a full-discrete spectralGalerkin exponential Euler approximation for the SPDE (7.76).
1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x ; b = @( x ) (1´x )./(1+ x . ˆ 2 ) / 4 ;5 x i = zeros (1 ,M) ;6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = ExpEuler (T, v , f , b , xi ,N,M) ;9 plot ( Preimage ,Y) ;
10 Y = ExpEuler (T, v , f , b , xi ,N,M) ;11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = ExpEuler (T, v , f , b , xi ,N,M) ;13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f
Matlab code 8.2: A Matlab code for simulating a full-discrete spectral Galerkinexponential Euler approximation for an SPDE of the form (7.76).
8.4. FULL DISCRETIZATIONS FOR SPDES 247
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 8.1: Result of a call of the Matlab code 8.2.
248 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
8.4.3 Full-discrete spectral Galerkin linear-implicit Euler methodfor SPDEs
Definition 8.4.2 (Full-discrete spectral Galerkin linear-implicit Euler approxima-tions). Assume the setting in Section 8.4.1, let N P N, I P PpBq, J P PpUq, andlet Y : t0, 1, . . . , Nu ˆΩ Ñ πIpHγq be an pFnTNqnPt0,1,...,Nu-adapted stochastic processwhich fulfills Y0 “ πIpξq and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holdsP-a.s. that
Yn`1 “`
IdH ´TNA˘´1
˜
Yn ` πI`
F pYnq˘
TN`
ż pn`1qTN
nTN
πI`
BpYnq$JpdWsq˘
¸
.
(8.145)
Then we call Y a full-discrete spectral Galerkin linear-implicit Euler approximationfor the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.146)
with time step size TN, spatial approximation I and noise approximation J .
1 function Y = LinImpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;6 y = y + f ( y )∗T/N + b( y ) . ∗dW;7 Y = i d s t ( y ) / sqrt (2 ) . / ( 1 ´ A∗T/N ) ;8 end9 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;
10 end
Matlab code 8.3: A Matlab function for simulating a full-discrete spectralGalerkin linear-implicit Euler approximation of the SPDE (7.76).
1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x ; b = @( x ) (1´x )./(1+ x . ˆ 2 ) / 4 ;5 x i = zeros (1 ,M) ;
8.4. FULL DISCRETIZATIONS FOR SPDES 249
6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = LinImpEuler (T, v , f , b , xi ,N,M)9 plot ( Preimage ,Y) ;
10 Y = LinImpEuler (T, v , f , b , xi ,N,M)11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = LinImpEuler (T, v , f , b , xi ,N,M)13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f
Matlab code 8.4: A Matlab code for simulating a full-discrete spectral Galerkinlinear-implicit Euler approximation of the SPDE (7.76).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 8.2: Result of a call of the Matlab code 8.4.
250 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
8.4.4 Full-discrete spectral Galerkin nonlinearity-stopped ex-ponential Euler method for SPDEs
Definition 8.4.3 (Full-discrete spectral Galerkin nonlinearity-stopped exponentialEuler approximations). Assume the setting in Section 8.4.1, let N P N, I P PpBq,J P PpUq, α P rγ ´ η, γs, assume that F pHγq Ď Hα and let Y : t0, 1, . . . , Nu ˆ Ω ÑπIpHγq be an pFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “ πIpξqand which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that
Yn`1 “ 1tπIpF pYnqq2HαąNTuYn (8.147)
` 1tπIpF pYnqq2HαďNTueA
TN
˜
Yn ` πI`
F pYnq˘
TN`
ż pn`1qTN
nTN
πI`
BpYnq$JpdWsq˘
¸
.
Then we call Y a full-discrete spectral Galerkin nonlinearity-stopped exponential Eu-ler approximation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.148)
with time step size TN, spatial approximation I and noise approximation J .
1 function Y = StopExpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 z = f ( y ) ;6 i f ( sum( i d s t ( z ) . ˆ 2 ) > 2∗N/T ) break ; end7 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;8 y = y + z∗T/N + b( y ) . ∗dW;9 Y = exp( A∗T/N ) .∗ i d s t ( y ) / sqrt ( 2 ) ;
10 end11 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;12 end
Matlab code 8.5: A Matlab function for simulating a full-discrete spectralGalerkin nonlinearity-stopped exponential Euler approximation for a generalized ver-sion of the SPDE (7.76) with γ P p1
5, 1
4q and α “ 0.
1 clear a l l2 rng ( ’ d e f a u l t ’ )
8.4. FULL DISCRETIZATIONS FOR SPDES 251
3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x . ˆ 3 ; b = @( x ) x /4 ;5 x i = zeros (1 ,M) ;6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = StopExpEuler (T, v , f , b , xi ,N,M) ;9 plot ( Preimage ,Y) ;
10 Y = StopExpEuler (T, v , f , b , xi ,N,M) ;11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = StopExpEuler (T, v , f , b , xi ,N,M) ;13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f
Matlab code 8.6: A Matlab code for simulating a full-discrete spectral Galerkinnonlinearity-stopped exponential Euler approximation for a generalized version ofthe SPDE (7.76) with γ P p1
5, 1
4q and α “ 0.
8.4.5 Full-discrete spectral Galerkin nonlinearity-stopped linear-implicit Euler method for SPDEs
Definition 8.4.4 (Full-discrete spectral Galerkin nonlinearity-stopped linear-im-plicit Euler approximations). Assume the setting in Section 8.4.1, let N P N, I PPpHq, J P PpUq, α P rγ´ η, γs, assume that F pHγq Ď Hα and let Y : t0, 1, . . . , NuˆΩ Ñ πIpHγq be an pFnTNqnPt0,1,...,Nu-adapted stochastic process which fulfills Y0 “
πIpξq and which fulfills that for all n P t0, 1, . . . , N ´ 1u it holds P-a.s. that
Yn`1 “ 1tπIpF pYnqq2HαąNTuYn (8.149)
` 1tπIpF pYnqq2HαďNTu
`
IdH ´TNA˘´1
˜
Yn ` πI`
F`
Yn˘˘
TN`
ż pn`1qTN
nTN
πI`
BpYnq$JpdWsq˘
¸
.
Then we call Y a spectral Galerkin nonlinearity-stopped linear-implicit Euler approx-imation for the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ (8.150)
with time step size TN, spatial approximation I and noise approximation J .
252 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
Figure 8.3: Result of a call of the Matlab code 8.6.
8.4. FULL DISCRETIZATIONS FOR SPDES 253
1 function Y = StopLinImpEuler (T, v , f , b , xi ,N,M)2 A = ´v∗pi ˆ2∗ (1 :M) . ˆ 2 ; Y = i d s t ( x i ) / sqrt ( 2 ) ;3 for n=1:N4 y = dst (Y) ∗ sqrt ( 2 ) ;5 z = f ( y ) ;6 i f ( sum( i d s t ( z ) . ˆ 2 ) > 2∗N/T ) break ; end7 dW = dst ( randn (1 ,M) .∗ sqrt (2∗T/N) ) ;8 y = y + z∗T/N + b( y ) . ∗dW;9 Y = i d s t ( y ) / sqrt (2 ) . / ( 1 ´ A∗T/N ) ;
10 end11 Y = [ 0 , dst (Y)∗ sqrt ( 2 ) , 0 ] ;12 end
Matlab code 8.7: A Matlab function for simulating a full-discrete spectralGalerkin nonlinearity-stopped linear-implicit Euler approximation for a generalizedversion of the SPDE (7.76) with γ P p1
5, 1
4q and α “ 0.
1 clear a l l2 rng ( ’ d e f a u l t ’ )3 T = 1 ; v = 1/50 ; M = 2ˆ8´1; N = Mˆ2 ;4 f = @( x ) 1´x . ˆ 3 ; b = @( x ) x /4 ;5 x i = zeros (1 ,M) ;6 Preimage = ( 0 :M+1)/(M+1);7 hold on8 Y = StopLinImpEuler (T, v , f , b , xi ,N,M) ;9 plot ( Preimage ,Y) ;
10 Y = StopLinImpEuler (T, v , f , b , xi ,N,M) ;11 plot ( Preimage ,Y, ’ r ’ ) ;12 Y = StopLinImpEuler (T, v , f , b , xi ,N,M) ;13 plot ( Preimage ,Y, ’ g ’ ) ;14 hold o f f
Matlab code 8.8: A Matlab code for simulating a spectral Galerkin nonlinearity-stopped linear-implicit Euler approximation for a generalized version of theSPDE (7.76) with γ P p1
5, 1
4q and α “ 0.
254 CHAPTER 8. STRONG NUMERICAL APPROXIMATIONS FOR SPDES
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
Figure 8.4: Result of a call of the Matlab code 8.8.
Chapter 9
Weak numerical approximationsfor SPDEs
9.1 An Ito type formula for SPDEs
Most of the following material comes from Da Prato et al. [6].
9.1.1 A setting for mild stochastic calculus
Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq,pH, 〈¨, ¨〉H , ¨Hq, pH, 〈¨, ¨〉H , ¨Hq, and pU, 〈¨, ¨〉U , ¨Uq be separableR-Hilbert spaces
with H Ď H Ď H continuously and densely, let pWtqtPr0,T s be a cylindrical IdU -Wiener process with respect to pFtqtPr0,T s, and let = Ď r0, T s2 be the set given by= “ tpt1, t2q P r0, T s
2 : t1 ă t2u.
255
256 CHAPTER 9. WEAK NUMERICAL APPROXIMATIONS FOR SPDES
9.1.2 Mild stochastic processes
Definition 9.1.1 (Mild Ito process). Assume the setting in Section 9.1.1, let S PM
`
Bp=q,BpLpH, Hqq˘
satisfy that for all t1, t2, t3 P r0, T s with t1 ă t2 ă t3 it
holds that St2,t3St1,t2 “ St1,t3, and let X : r0, T s ˆ Ω Ñ H, Y : r0, T s ˆ Ω Ñ H, and
Z : r0, T sˆΩ Ñ HSpU, Hq be pFtqtPr0,T s-predictable stochastic processes which satisfy
that for all t P r0, T s it holds P-a.s. thatşt
0Ss,tYsH ` Ss,tZs
2HSpU,Hq ds ă 8 and
which satisfy that for all t P p0, T s it holds P-a.s. that
Xt “ S0,tX0 `
ż t
0
Ss,t Ys ds`
ż t
0
Ss,t Zs dWs. (9.1)
Then we call X a mild Ito process (with evolution family S, mild drift Y , and milddiffusion Z).
Note that if pH, 〈¨, ¨〉H , ¨Hq “ pH, 〈¨, ¨〉H , ¨Hq and if the evolution family
S : = Ñ LpHq satisfies @ pt1, t2q P = : St1,t2 “ IdH in Definition 9.1.1, then themild Ito process (9.1) is nothing else but a “standard” Ito process. (In the followingthe terminology “standard Ito process” instead of simply “Ito process” is used inorder to distinguish more clearly from the above notion of a mild Ito process.) Everystandard Ito process is thus also a mild Ito process. However, a mild Ito processis, in general, not a standard Ito process. The concept of a mild Ito process inDefinition 9.1.1 thus generalizes the concept of a standard Ito process. In concreteexamples of mild Ito processes it will be crucial that the semigroup S : = Ñ LpH, Hqin Definition 9.1.1 depends explicitly on both variables t1 and t2 with pt1, t2q P =
instead of on t2 ´ t1 only (cf. Section 8.2). The assumption that for all t P r0, T sit holds P-a.s. that
şt
0Ss,tYsH ` Ss,tZs
2HSpU,Hq ds ă 8 in Definition 9.1.1 ensures
that both the deterministic and the stochastic integral in (9.1) are well defined. Inthe next step an immediate consequence of Definition 9.1.1 is presented.
Proposition 9.1.2. Assume the setting in Section 9.1.1 and let X : r0, T sˆΩ Ñ H bea mild Ito process with evolution family S : = Ñ LpH, Hq, mild drift Y : r0, T sˆΩ ÑH, and mild diffusion Z : r0, T s ˆ Ω Ñ HSpU, Hq. Then for all t1, t2 P r0, T s witht1 ă t2 it holds P-a.s. that
Xt2 “ St1,t2 Xt1 `
ż t2
t1
Ss,t2 Ys ds`
ż t2
t1
Ss,t2 Zs dWs. (9.2)
9.1. AN ITO TYPE FORMULA FOR SPDES 257
Obviously, equation (9.2) in Proposition 9.1.2 generalizes equation (9.1) in thedefinition of a mild Ito process. Let us complete this subsection on mild Ito processeswith the notion of a certain subclass of mild Ito processes.
Definition 9.1.3 (Mild martingale). Assume the setting in Section 9.1.1 and letX : r0, T sˆΩ Ñ H be a mild Ito process with evolution family S : = Ñ LpH, Hq, milddrift Y : r0, T sˆΩ Ñ H, and mild diffusion Z : r0, T sˆΩ Ñ HSpU, Hq satisfying thatfor all t P r0, T s it holds that E
“
XtH
‰
ă 8 and satisfying that for all t1, t2 P r0, T swith t1 ă t2 it holds P-a.s. that
E“
Xt2
ˇ
ˇFt1‰
“ St1,t2 Xt1 . (9.3)
Then we call X a mild martingale (with respect to the filtration pFtqtPr0,T s and withrespect to the semigroup S).
9.1.3 Mild Ito formula
Theorem 9.1.4 (Mild Ito formula). Assume the setting in Section 9.1.1, let U Ď Ube an orthonormal basis of U , let pV, 〈¨, ¨〉V , ¨V q be a separable R-Hilbert space, letϕ “ pϕpr, xqqrPr0,T s,xPH P C
1,2pr0, T s ˆ H, V q, and let X : r0, T s ˆ Ω Ñ H be a mild
Ito process with evolution family S : = Ñ LpH, Hq, mild drift Y : r0, T s ˆ Ω Ñ H,and mild diffusion Z : r0, T s ˆ Ω Ñ HSpU, Hq. Then for all t0, t P r0, T s with t0 ă tit holds P-a.s. that
ż t
t0
›
›
`
B
Bxϕ˘
ps, Ss,tXsqSs,tYs›
›
V`›
›
`
B
Bxϕqps, Ss,tXsqSs,tZs
›
›
2
HSpU,V qds ă 8, (9.4)
ż t
t0
›
›
`
B
Brϕ˘
ps, Ss,tXsq›
›
V`
`
B2
Bx2ϕ˘
ps, Ss,tXsqLp2qpH,V q Ss,tZs2HSpU,Hq
ds ă 8, (9.5)
ϕpt,Xtq “ ϕpt0, St0,tXt0q `t
∫t0
`
B
Brϕ˘
ps, Ss,tXsq ds`t
∫t0
`
B
Bxϕ˘
ps, Ss,tXsqSs,t Ys ds
`
ż t
t0
`
B
Bxϕ˘
ps, Ss,tXsqSs,t Zs dWs `12
ř
uPU
tş
t0
`
B2
Bx2ϕ˘
ps, Ss,tXsq pSs,tZsu, Ss,tZsuq ds.
(9.6)
258 CHAPTER 9. WEAK NUMERICAL APPROXIMATIONS FOR SPDES
Note that (9.4) and (9.5) ensure that the possibly infinite sum and all integralsin (9.6) are well defined. Indeed, equation (9.5) implies that for all t0, t P r0, T s witht0 ă t it holds P-a.s. that
ÿ
uPU
ż t
t0
›
›
`
B2
Bx2ϕ˘
ps, Ss,tXsqpSs,tZsu, Ss,tZsuq›
›
Vds
ď
ż t
t0
›
›
`
B2
Bx2ϕ˘
ps, Ss,tXsq›
›
Lp2qpH,V q
´
ÿ
uPUSs,tZsu
2H
¯
ds
“
ż t
t0
›
›
`
B2
Bx2ϕ˘
ps, Ss,tXsq›
›
Lp2qpH,V q
›
›Ss,tZs›
›
2
HSpU,Hqds ă 8.
(9.7)
Moreover, note that the mild Ito formula (9.6) is independent of the particular choiceof the orthonormal basis U Ď U of U . If the test function pϕpt, xqqtPr0,T s,xPH P
C1,2pr0, T s ˆ H, V q in the mild Ito formula (9.6) does not depend on t P r0, T s, thenthe mild Ito formula in Theorem 9.1.4 reads as follows.
Corollary 9.1.5. Assume the setting in Section 9.1.1, let U Ď U be an orthonormalbasis of U , let pV, 〈¨, ¨〉V , ¨V q be a separable R-Hilbert space, let ϕ P C2pH, V q, and
let X : r0, T s ˆΩ Ñ H be a mild Ito process with evolution family S : = Ñ LpH, Hq,mild drift Y : r0, T s ˆ Ω Ñ H, and mild diffusion Z : r0, T s ˆ Ω Ñ HSpU, Hq. Thenfor all t0, t P r0, T s with t0 ă t it holds P-a.s. that
ż t
t0
ϕ1pSs,tXsqSs,tYsV ` ϕ1pSs,tXsqSs,tZs
2
HSpU,V q ds ă 8, (9.8)
ż t
t0
ϕ2pSs,tXsqLp2qpH,V q Ss,tZs2HSpU,Hq
ds ă 8, (9.9)
ϕpXtq “ ϕpSt0,tXt0q `
ż t
t0
ϕ1pSs,tXsqSs,t Ys ds`
ż t
t0
ϕ1pSs,tXsqSs,t Zs dWs
`1
2
ÿ
uPU
ż t
t0
ϕ2pSs,tXsq pSs,tZsu, Ss,tZsuq ds. (9.10)
Corollary 9.1.5 is an immediate consequence of Theorem 9.1.4. In our proof ofTheorem 9.1.4 below the following elementary result is used.
Exercise 9.1.6. Let Y, Z : r0, T s ˆ Ω Ñ r0,8q be two product measurable stochas-tic processes with the property that for all t P r0, T s it holds that P
“
Yt “ Zt‰
“
P“ şT
0Ys ds ă 8
‰
“ 1. Then P“ şT
0Zs ds ă 8
‰
“ 1.
9.1. AN ITO TYPE FORMULA FOR SPDES 259
In the next step the proof of Theorem 9.1.4 is presented (cf. Conus & Dalang [4],Conus [3], Lindner & Schilling [22], Kovacs, Larsson & Lindgren [19], Debussche &Printemps [8]).
Proof of Theorem 9.1.4. For every t P p0, T s let X t : r0, ts ˆΩ Ñ H be an pFsqsPr0,ts-adapted stochastic processes with continuous sample paths satisfying that for allu P r0, ts, t P p0, T s it holds P-a.s. that
X tu “ S0,tX0 `
ż u
0
Ss,t Ys ds`
ż u
0
Ss,t Zs dWs. (9.11)
Note that the assumption that for all t P r0, T s it holds P-a.s. thatşt
0Ss,tYsH `
Ss,tZs2HSpU,Hq ds ă 8 (see Definition 9.1.1) ensures for all t P p0, T s that X t : rτ, tsˆ
Ω Ñ H in (9.11) is indeed a well defined pFsqsPr0,ts-adapted stochastic processes withcontinuous sample paths. In the next step the continuity of the partial derivativesof ϕ : r0, T s ˆ H Ñ V , the continuity of the sample paths of X t : rτ, ts ˆ Ω Ñ H,t P p0, T s, and again the assumption that for all t P r0, T s it holds P-a.s. thatşt
0Ss,tYsH ` Ss,tZs
2HSpU,Hq ds ă 8 imply that for all t0, t P r0, T s with t0 ă t it
holds that
P„ż t
t0
›
›
`
B
Bxϕ˘
ps, X tsqSs,tYs
›
›
V`›
›
`
B
Bxϕ˘
ps, X tsqSs,tZs
›
›
2
HSpU,V qds ă 8
“ 1 (9.12)
and
P„ż t
t0
›
›
`
B
Brϕ˘
ps, X tsq›
›
V`›
›
`
B2
Bx2ϕ˘
ps, X tsq›
›
Lp2qpH,V q
›
›Ss,tZs›
›
2
HSpU,Hqds ă 8
“ 1.
(9.13)Moreover, the fact that for all t P p0, T s it holds P-a.s. that Xt “ X t
t and the standardIto formula (see Theorem 2.4 in Brzezniak, Van Neerven, Veraar & Weis [1]) provethat for all t0, t P r0, T s with t0 ă t it holds P-a.s. that
ϕpt,Xtq “ ϕpt, X tt q “ ϕpt0, X
tt0q `
ż t
t0
`
B
Brϕ˘
ps, X tsq ds`
ż t
t0
`
B
Bxϕ˘
ps, X tsqSs,t Ys ds
`
ż t
t0
`
B
Bxϕ˘
ps, X tsqSs,t Zs dWs `
1
2
ÿ
uPU
ż t
t0
`
B2
Bx2ϕ˘
ps, X tsq pSs,t Zs u, Ss,t Zs uq ds.
(9.14)
260 CHAPTER 9. WEAK NUMERICAL APPROXIMATIONS FOR SPDES
Moreover, note that for all s, t P r0, T s with s ă t it holds P-a.s. that
X ts “ S0,tX0 `
ż s
0
Su,t Yu du`
ż s
0
Su,t Zu dWu
“ Ss,t
ˆ
S0,sX0 `
ż s
0
Su,s Yu du`
ż s
0
Su,s Zu dWu
˙
“ Ss,tXs
(9.15)
Combining (9.15), (9.12), (9.13) with Exercise 9.1.6 implies (9.4) and (9.5). More-over, putting (9.15) into (9.14) proves (9.6). The proof of Theorem 9.1.4 is thuscompleted.
9.2 Solution processes of SPDEs
A direct consequence of Theorem 9.1.4 and Corollary 9.1.5 is the next corollary.
Corollary 9.2.1 (Mild Ito formula for solutions of SPDEs). Assume the setting inSection 7.1.2, let U Ď U be an orthonormal basis of U , and let X : r0, T sˆΩ Ñ Hγ bethe up to modifications unique pFtqtPr0,T s-predictable stochastic process which satisfiessuptPr0,T s XtLppP;¨Hγ q
ă 8 and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (9.16)
Then for all r P p´8,mintγ´η`1, β` 12uq, ϕ P C2pHr, V q, t0, t P r0, T s with t0 ă tit holds P-a.s. that
ż t
t0
›
›ϕ1peApt´sqXsq eApt´sqF pXsq
›
›
Vds ă 8, (9.17)
ż t
t0
›
›ϕ1peApt´sqXsq eApt´sqBpXsq
›
›
2
HSpU,V qds ă 8, (9.18)
ż t
t0
›
›ϕ2peApt´sqXsq›
›
Lp2qpHr,V q
›
›eApt´sqBpXsq›
›
2
HSpU,Hrqds ă 8, (9.19)
ϕpXtq “ ϕpeApt´t0qXt0q `
ż t
t0
ϕ1peApt´sqXsq eApt´sqF pXsq ds
`
ż t
t0
ϕ1peApt´sqXsq eApt´sqBpXsq dWs (9.20)
`1
2
ÿ
uPU
ż t
t0
ϕ2peApt´sqXsq`
eApt´sqBpXsqu, eApt´sqBpXsqu
˘
ds.
9.3. TRANSFORMATIONS OF SEMIGROUPS OF SOLUTIONS OF SPDES 261
9.3 Transformations of semigroups of solutions of
SPDEs
In our definition of a mild solution process in Definition 7.1.1 above we assume thatsuppσP pAqq ă 8. For symmetric diagonal linear operators A : DpAq Ď H Ñ H thecondition suppσP pAqq ă 8 is equivalent to the condition that A is the generator of astrongly continuous semigroup; see Proposition 4.7.2 above for details. In our settingin Section 7.1.2 we impose the more restricitive condition that suppσP pAqq ă 0. Thenext proposition illustrates that the more restricitive condition suppσP pAqq ă 0 doesin a suitable sense not reduce the generality.
Proposition 9.3.1. Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be two separable R-Hilbert spaces, let A : DpAq Ď H Ñ H be a symmetric diagonal linear operator withsuppσP pAqq ă 8, let η P psuppσP pAqq,8q, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a familyof interpolation spaces associated to η ´A, let T P r0,8q, α, β, γ, ι P R, O P BpHγq,F P M
`
BpOq,BpHαq˘
, F P M`
BpOq,BpHminpα,γqq˘
, B P M`
BpOq,BpHSpU,Hβqq˘
fulfill that for all v P O it holds that F pvq “ ιv ` F pvq, let pΩ,F ,P, pFtqtPr0,T sq bea stochastic basis, let ξ P MpF0,BpOqq, let pWtqtPr0,T s be a cylindrical IdU -Wienerprocess w.r.t. pFtqtPr0,T s, and let X : r0, T s ˆΩ Ñ O be a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ, (9.21)
i.e., assume that X is an pFtqtPr0,T s-predictable stochastic process which fulfills thatfor all t P r0, T s it holds P-a.s. that
ż t
0
eApt´sqF pXsqHγ ` eApt´sqBpXsq
2HSpU,Hγq ds ă 8 (9.22)
and Xt “ eAtξ `
ż t
0
eApt´sqF pXsq ds`
ż t
0
eApt´sqBpXsq dWs. (9.23)
Then X is a mild solution of the SPDE
dXt ““
pA´ ιqXt ` F pXtq‰
dt`BpXtq dWt, t P r0, T s, X0 “ ξ, (9.24)
i.e., for all t P r0, T s it holds P-a.s. thatż t
0
epA´ιqpt´sqF pXsqHγ ` epA´ιqpt´sqBpXsq
2HSpU,Hγq ds ă 8 (9.25)
and Xt “ epA´ιqtξ `
ż t
0
epA´ιqpt´sqF pXsq ds`
ż t
0
epA´ιqpt´sqBpXsq dWs. (9.26)
262 CHAPTER 9. WEAK NUMERICAL APPROXIMATIONS FOR SPDES
Proof of Proposition 9.3.1. First of all, observe that (9.22)–(9.23) imply that for allt P r0, T s it holds P-a.s. that
ż t
0
epA´ιqpt´sq e´ιsF pXsqHγ ` epA´ιqpt´sq e´ιsBpXsq
2HSpU,Hγq ds
“ e´ιtż t
0
eApt´sqF pXsqHγ ` eApt´sqBpXsq
2HSpU,Hγq ds ă 8 and
(9.27)
e´ιtXt “ epA´ιqtξ `
ż t
0
epA´ιqpt´sq e´ιsF pXsq ds`
ż t
0
epA´ιqpt´sq e´ιsBpXsq dWs. (9.28)
This implies that the stochastic process pe´ιtXtqtPr0,T s is a mild Ito process withevolution family epA´ιqpt´sq, ps, tq P tpt1, t2q P r0, T s
2 : t1 ă t2u, mild drift e´ιsF pXsq,s P r0, T s, and mild diffusion e´ιsBpXsq, s P r0, T s. Next let ψ, ψ1 : r0, T sˆHγ Ñ Hγ
and ψ2 : r0, T s ˆHγ Ñ LpHγq be the functions with the property that for all pt, xq Pr0, T s ˆHγ, v P Hγ it holds that
ψpt, xq “ eιtx, ψ1pt, xq “B
Btψpt, xq “ ι ψpt, xq, ψ2pt, xq v “
B
Bxψpt, xq v “ ψpt, vq.
(9.29)The mild Ito formula in Theorem 9.1.4 then proves that for all t P r0, T s it holdsP-a.s. that
ż t
0
›
›ψ1
`
s, epA´ιqpt´sq e´ιsXs
˘
loooooooooooooomoooooooooooooon
“ι epA´ιqpt´sqXs
›
›
Hγds ă 8, (9.30)
ż t
0
›
›ψ2
`
s, epA´ιqpt´sq e´ιsXs
˘
epA´ιqpt´sq e´ιsF pXsqloooooooooooooooooooooooooooomoooooooooooooooooooooooooooon
“epA´ιqpt´sqF pXsq
›
›
Hγds ă 8, (9.31)
ż t
0
›
›ψ2
`
s, epA´ιqpt´sq e´ιsXs
˘
epA´ιqpt´sq e´ιsBpXsqloooooooooooooooooooooooooooomoooooooooooooooooooooooooooon
“epA´ιqpt´sqBpXsq
›
›
2
HSpU,Hγqds ă 8, and (9.32)
Xt “ ψ`
t, e´ιtXt
˘
“ ψ`
0, epA´ιqt e´ι0X0
˘
looooooooooomooooooooooon
“epA´ιqtX0
`
ż t
0
epA´ιqpt´sq rιXs ` F pXsqs ds`
ż t
0
epA´ιqpt´sqBpXsq dWs.
(9.33)
The proof of Proposition 9.3.1 is thus completed.
9.4. WEAK CONVERGENCE FOR TEMPORAL NUMERICAL APPROXIMATIONS FOR SPDES263
9.4 Weak convergence for temporal numerical ap-
proximations for SPDEs
Exercise 9.4.1. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let T P
p0,8q, N P N, ϕ : L2pBorelp0,1q; |¨|Rq Ñ R satisfy that for all v P L2pBorelp0,1q; |¨|Rq
it holds that ϕpvq “ v2L2pBorelp0,1q;|¨|Rq, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic ba-
sis, let pWtqtPr0,T s be a cylindrical IdL2pBorelp0,1q;|¨|Rq-Wiener process w.r.t. pFtqtPr0,T s,
let Y : r0, T s ˆ Ω Ñ L2pBorelp0,1q; |¨|Rq be a naturally-interpolated exponential Eulerapproximation for the SPDE
dXt “ AXt dt` dWt, t P r0, T s, X0 “ 0 (9.34)
with time step size TN, and let X : r0, T s ˆ Ω Ñ L2pBorelp0,1q; |¨|Rq be an pFtqtPr0,T s-predictable stochastic process which fulfills that for all t P r0, T s it holds P-a.s. that
Xt “
ż t
0
eApt´sq dWs. (9.35)
Prove that for all r P“
0, 12˘
it holds thatˇ
ˇE“
ϕpXT q‰
´E“
ϕpYT q‰ˇ
ˇ ď T r
Nr p12´rq.
We refer to [15] and the references mentioned therein for further weak convergenceresults for SPDEs.
9.5 Weak convergence of Galerkin projections for
SPDEs
The following material comes from Section 2 in Conus et al. [5].
9.5.1 Setting
Let pH, 〈¨, ¨〉H , ¨Hq and pU, 〈¨, ¨〉U , ¨Uq be separable R-Hilbert spaces, let T P
p0,8q, η P r0, 1q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s be acylindrical IdU -Wiener process with respect to pFtqtPr0,T s, let H Ď H be an orthonor-mal basis of H, let λ : HÑ R be a function satisfying supbPH λb ă 0, let A : DpAq ĎH Ñ H be a linear operator such that DpAq “ tv P H :
ř
bPH |λb 〈b, v〉H |2ă 8u and
such that for all v P DpAq it holds that Av “ř
bPH λb 〈b, v〉H b, let pHr, 〈¨, ¨〉Hr , ¨Hrq,r P R, be a family of interpolation spaces associated to ´A, and let F P C0,1pH,H´ηq,
264 CHAPTER 9. WEAK NUMERICAL APPROXIMATIONS FOR SPDES
B P C0,1pH,HSpU,H´η2qq, ϕ P C2pH,Rq, ξ P L3pP|F0 ; ¨Hq, pπIqIPPpHq Ď LpH´ηq
fulfill that for all v P H, I P PpHq it holds that πIpvq “ř
bPI 〈b, v〉H b.The above assumptions ensure (see Theorem 7.1.9) that there exist an up to
modifications unique pFtqtPr0,T s-predictable stochastic process X : r0, T s ˆ Ω Ñ Hwhich satisfies suptPr0,T s XtL3pP;¨Hq
ă 8 and which satisfies that for all t P r0, T s itholds P-a.s. that
Xt “ eAtξ `
ż t
0
eApt´sqF pXsq ds`
ż t
0
eApt´sqBpXsq dWs. (9.36)
9.5.2 Weak convergence for spatial spectral Galerkin pro-jections
Proposition 9.5.1. Assume the setting in Section 9.5.1 and let ρ P r0, 1 ´ ηq,I P PpHq. Then
ˇ
ˇE“
ϕpXT q‰
´E“
ϕ`
πIpXT q˘‰ˇ
ˇ
ď“
ϕLippH,Rq ` |ϕ1|C0,1pH,LpH,Rqq ` |ϕ
2|C0,1pH,Lp2qpH,Rqq
‰
max
1, suptPr0,T sE“
Xt3H
‰(
¨
«
1
T ρ`T p1´ρ´ηq
“
F LippH,H´ηq ` B2LippH,HSpU,H´η2qq
‰
p1´ ρ´ ηq
ff
“
infbPHzI |λb|‰´ρ
. (9.37)
Proof. Throughout this proof let U Ď U be an orthonormal basis of U and letBb P CpH,H´η2q, b P U, be the functions with the property that for all v P H,b P U it holds that Bbpvq “ Bpvq b. Then observe that the mild Ito formula inCorollary 9.2.1 yields that
E“
ϕpXT q‰
´E“
ϕpπIpXT qq‰
“ E“
ϕpeAT ξq‰
´E“
ϕpeATπIpξqq‰
`
ż T
0
E“
ϕ1peApT´tqXtq eApT´tqF pXtq
‰
dt
´
ż T
0
E“
ϕ1peApT´tqπIpXtqq eApT´tqπIF pXtq
‰
dt
`1
2
ÿ
bPU
ż T
0
E“
ϕ2peApT´tqXtqpeApT´tqBb
pXtq, eApT´tqBb
pXtqq‰
dt
´1
2
ÿ
bPU
ż T
0
E“
ϕ2peApT´tqπIpXtqqpeApT´tqπIB
bpXtq, e
ApT´tqπIBbpXtqq
‰
dt.
(9.38)
9.5. WEAK CONVERGENCE OF GALERKIN PROJECTIONS FOR SPDES265
Next observe that the fact that for all r P r0,8q it holds that suptPp0,8q›
›p´tAqreAt›
›
LpHqď
supxPp0,8q“
xr
ex
‰
ď“
re
‰r(see Theorem 4.7.6) implies that
ˇ
ˇE“
ϕpeAT ξq‰
´E“
ϕpeATπIpξqq‰ˇ
ˇ ď|ϕ|C0,1pH,Rq ξL1pP;¨Hq
πHzILpH,H´ρq
T ρ. (9.39)
Inequality (9.39) provides us a bound for the first difference on the right hand sideof (9.38). In the next step we bound the second difference on the right hand side of(9.38). For this observe that for all x P H, t P r0, T q it holds that
ˇ
ˇ
“
ϕ1peApT´tqxq ´ ϕ1peApT´tqπIpxqq‰
eApT´tqF pxqˇ
ˇ
ď|ϕ1|C0,1pH,LpH,Rq πHzILpH,H´ρq xH F pxqH´η
pT ´ tqpρ`ηq(9.40)
andˇ
ˇϕ1peApT´tqπIpxqq`
rIdH ´πIs eApT´tqF pxq
˘ˇ
ˇ
ď|ϕ|C0,1pH,Rq πHzILpH,H´ρq F pxqH´η
pT ´ tqpρ`ηq.
(9.41)
Combining (9.40) and (9.41) proves that
ˇ
ˇ
ˇ
ˇ
ż T
0
E“
ϕ1peApT´tqXtq eApT´tqF pXtq
‰
dt´
ż T
0
E“
ϕ1peApT´tqπIpXtqq eApT´tqπIF pXtq
‰
dt
ˇ
ˇ
ˇ
ˇ
ďT p1´ρ´ηq suptPr0,T sE
“
XtH F pXtqH´η |ϕ1|C0,1pH,LpH,Rqq`F pXtqH´η |ϕ|C0,1pH,Rq
‰
πHzILpH,H´ρq
p1´ρ´ηq
ď
T p1´ρ´ηq r|ϕ1|C0,1pH,LpH,Rqq`|ϕ|C0,1pH,Rqs suptPr0,T s
maxtErXtH F pXtqH´η s,ErF pXtqH´η su πHzILpH,H´ρq
p1´ρ´ηq
ďT p1´ρ´ηq r|ϕ1|C0,1pH,LpH,Rqq`|ϕ|C0,1pH,Rqs F LippH,H´ηq
maxt1,suptPr0,T sErXt2H su πHzILpH,H´ρq
p1´ρ´ηq.
(9.42)
Inequality (9.42) provides us a bound for the second difference on the right hand sideof (9.38). Next we bound the third difference on the right hand side of (9.38). Tothis end note that for all x P H, t P r0, T q it holds that
ˇ
ˇ
ˇ
ˇ
ř
bPU
“
ϕ2peApT´tqxq ´ ϕ2peApT´tqπIpxqq‰
peApT´tqBbpxq, eApT´tqBbpxqq
ˇ
ˇ
ˇ
ˇ
ď|ϕ2|C0,1pH,Lp2qpH,Rqq Bpxq
2HSpU,H´η2q
xH πHzILpH,H´ρq
pT ´ tqpρ`ηq
(9.43)
266 CHAPTER 9. WEAK NUMERICAL APPROXIMATIONS FOR SPDES
andˇ
ˇ
ˇ
ˇ
ř
bPUϕ2peApT´tqπIpxqqprIdH `πIse
ApT´tqBbpxq, rIdH ´πIseApT´tqBbpxqq
ˇ
ˇ
ˇ
ˇ
ď2 |ϕ1|C0,1pH,LpH,Rqq Bpxq
2HSpU,H´η2q
πHzILpH,H´ρq
pT ´ tqpρ`ηq.
(9.44)
Combining (9.43) and (9.44) proves that
ˇ
ˇ
ˇ
ˇ
ˇ
1
2
ÿ
bPU
ż T
0
E“
ϕ2peApT´tqXtqpeApT´tqBb
pXtq, eApT´tqBb
pXtqq‰
dt
´1
2
ÿ
bPU
ż T
0
E“
ϕ2peApT´tqπIpXtqqpeApT´tqπIB
bpXtq, e
ApT´tqπIBbpXtqq
‰
dt
ˇ
ˇ
ˇ
ˇ
ˇ
ďT p1´ρ´ηq πHzILpH,H´ρq
“
|ϕ2|C0,1pH,Lp2qpH,Rqq ` |ϕ1|C0,1pH,LpH,Rqq
‰
p1´ ρ´ ηq
¨
«
suptPr0,T s
max
E“
XtHBpXtq2HSpU,H´η2q
‰
,E“
BpXtq2HSpU,H´η2q
‰(
ff
ďT p1´ρ´ηq πHzILpH,H´ρq
“
|ϕ2|C0,1pH,Lp2qpH,Rqq ` |ϕ1|C0,1pH,LpH,Rqq
‰
p1´ ρ´ ηq
¨ B2LippH,HSpU,H´η2qqmax
!
1, suptPr0,T s
E“
Xt3H
‰
)
.
(9.45)
Combining (9.38), (9.39), (9.42), and (9.45) completes the proof of Proposition 9.5.1.
Question 9.5.2. Assume the setting in Section 9.5.1, assume that ϕ P C3pH,Rq,and assume that ϕ has globally bounded derivatives. For which ρ P R does it holdthat there exists a real number C P R such that for all I P PpHq it holds that
ˇ
ˇE“
ϕpXT q‰
´E“
ϕ`
πIpXT q˘‰ˇ
ˇ ď C“
infbPHzI |λb|‰´ρ
. (9.46)
9.5. WEAK CONVERGENCE OF GALERKIN PROJECTIONS FOR SPDES267
Question 9.5.3 (Weak convergence rates of spectral Galerkin approximations).Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pH, 〈¨, ¨〉H , ¨Hq“ pL2pBorelp0,1q; |¨|Rq, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq
, ¨L2pBorelp0,1q;|¨|Rqq, let pWtqtPr0,T s be a cylin-
drical IdH-Wiener process w.r.t. pFtqtPr0,T s, let b : p0, 1qˆRÑ R be a globally Lipschitzcontinuous function, let ξ P H, let X : r0, T s ˆ Ω Ñ H be a mild solution process ofthe SPDE
dXtpxq “B2
Bx2Xtpxq dt` bpx,Xtpxqq dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ ξpxq
(9.47)for x P p0, 1q, t P r0, T s, and let πN P LpHq, N P N, satisfy that for all N P N, v P Hit holds that
πNpvq “Nÿ
n“1
2 sinpnπp¨qq
ż 1
0
sinpnπxq vpxq dx. (9.48)
(i) For which r P R does it holds that there exists a real number C P r0,8q suchthat for all N P N it holds that ErXT ´ πNpXT qHs ď CN´r?
(ii) For which r P R does it holds that for every ϕ P C3pH,Rq with globally boundedderivatives there exists a real number C P r0,8q such that for all N P N it holdsthat
“
E“
ϕpXT q‰
´E“
ϕpπNpXT qq‰ˇ
ˇ ď CN´r?
268 CHAPTER 9. WEAK NUMERICAL APPROXIMATIONS FOR SPDES
Chapter 10
Additional material
10.1 Egorov’s theorem on almost uniform conver-
gence
See, e.g., Wikipedia http://en.wikipedia.org/wiki/Egorov%27s_theorem#Generalizationsfor the next results and their proofs.
10.1.1 General measure spaces
10.1.1.1 Almost sure convergence
Proposition 10.1.1 (A characterization for almost sure convergence). Let pΩ,F , µqbe a measure space, let pE, dq be a metric space, and let fn : Ω Ñ E, n P N0,be strongly F/pE, dq-measurable functions. Then the following two statements areequivalent:
(i) It holds µ-a.s. that limnÑ8 dpfn, f0q “ 0.
(ii) It holds for all ε P p0,8q that µ`
XnPNtsupmPNXrn,8q dpfm, f0q ą εu˘
“ 0.
Proof of Proposition 10.1.1. Observe that
µpΩz tlim supnÑ8 dpfn, f0q “ 0uq
“ µ´
Ω z
@ k P N : Dn P N : @m P NX rn,8q : dpfm, f0q ď1k
(
¯
“ µ´
Ω z
@ k P N : Dn P N : supmPNXrn,8q dpfm, f0q ď1k
(
¯
“ µ´
YkPN XnPN
supmPNXrn,8q dpfm, f0q ą1k
(
¯
.
(10.1)
269
270 CHAPTER 10. ADDITIONAL MATERIAL
This proves that it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 if and only if for all k P Nit holds that µ
`
XnPNtsupmPNXrn,8q dpfm, f0q ą1ku˘
“ 0. This completes the proof ofProposition 10.1.1.
Corollary 10.1.2 (Almost sure convergence). Let pΩ,F , µq be a measure space, letpE, dq be a metric space, and let fn : Ω Ñ E, n P N0, be strongly F/pE, dq-measurablefunctions with the property that it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0. Then itholds for all ε P p0,8q, A P F with µpAq ă 8 that
limnÑ8
µ`
AX tsupmPNXrn,8q dpfm, f0q ą εu˘
“ 0. (10.2)
Proof of Corollary 10.1.2. Proposition 10.1.1 implies that for all ε P p0,8q it holdsthat µ
`
XnPNtsupmPNXrn,8q dpfm, f0q ą εu˘
“ 0. This and continuity from above, inparticular, ensure that for all ε P p0,8q, A P F with µpAq ă 8 it holds that
0 “ µ`
AX`
XnPN tsupmPNXrn,8q dpfm, f0q ą εu˘˘
“ µ`
XnPN`
AX tsupmPNXrn,8q dpfm, f0q ą εu˘˘
“ limnÑ8
µ`
AX tsupmPNXrn,8q dpfm, f0q ą εu˘
.
(10.3)
The proof of Corollary 10.1.2 is thus completed.
Proposition 10.1.3 (A modified verison of the Sevirini-Egorov theorem). Let pΩ,F , µqbe a measure space, let pE, dq be a metric space, and let fn : Ω Ñ E, n P N0,be strongly F/pE, dq-measurable functions. Then the following two statements areequivalent:
(i) It holds for all ε P p0,8q, A P F that there exist a set B P pA \ Fq such thatµpAzBq ă ε and limnÑ8 supωPB dpfnpωq, f0pωqq “ 0.
(ii) It holds for all ε P p0,8q, A P F with µpAq ă 8 that
limnÑ8
µ`
AX tsupmPNXrn,8q dpfm, f0q ą εu˘
“ 0. (10.4)
Proof of Proposition 10.1.3. Proposition 10.1.3 implies that for all ε P p0,8q it holdsthat µ
`
XnPNtsupmPNXrn,8q dpfm, f0q ą εu˘
“ 0. This and continuity from above, inparticular, ensure that for all ε P p0,8q, A P F with µpAq ă 8 it holds that
0 “ µ`
AX`
XnPN tsupmPNXrn,8q dpfm, f0q ą εu˘˘
“ µ`
XnPN`
AX tsupmPNXrn,8q dpfm, f0q ą εu˘˘
“ limnÑ8
µ`
AX tsupmPNXrn,8q dpfm, f0q ą εu˘
.
(10.5)
The proof of Proposition 10.1.3 is thus completed.
10.1. EGOROV’S THEOREM ON ALMOST UNIFORM CONVERGENCE 271
10.1.1.2 Luzin uniform-type convergence
Lemma 10.1.4 (Luzin uniform-type convergence implies almost sure convergence).Let pΩ,F , µq be a measure space, let pE, dq be a metric space, and let fn : Ω Ñ E,n P N0, be strongly F/pE, dq-measurable functions with the property that there existsets Ak P F , k P N, such that @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 andµ`
ΩzpYkPNAkq˘
“ 0. Then it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0.
Proof of Lemma 10.1.4. Observe that the assumption that for all k P N it holds thatlimnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 implies that
pYkPNAkq Ď!
ω P Ω: limnÑ8
dpfnpωq, f0pωqq “ 0)
. (10.6)
Combining this with the assumption that µ`
ΩzpYkPNAkq˘
“ 0 completes the proofof Lemma 10.1.4.
10.1.1.3 Almost uniform convergence
Lemma 10.1.5 (Almost uniform convergence implies Luzin uniform-type conver-gence). Let pΩ,F , µq be a measure space, let pE, dq be a metric space, and let fn : Ω ÑE, n P N0, be strongly F/pE, dq-measurable functions with the property that for allε P p0,8q there exists a set A P F such that limnÑ8 supωPA dpfnpωq, f0pωqq “ 0and µpΩzAq ă ε. Then it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 and it holdsthat there exist sets Ak P F , k P N, such that µ
`
ΩzpYkPNAkq˘
“ 0 and @ k PN : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0.
Proof of Lemma 10.1.5. By assumption that there exists sets Ak P F , k P N, suchthat for all k P N it holds that
limnÑ8
supωPAk
dpfnpωq, f0pωqq “ 0 and µpΩzAkq ă1k. (10.7)
This ensures that
µpΩz pYkPNAkqq “ µpXkPN pΩzAkqq ě limkÑ8
µpΩzAkq “ 0. (10.8)
Combining this and (10.7) with Lemma 10.1.4 completes the proof of Lemma 10.1.5.
272 CHAPTER 10. ADDITIONAL MATERIAL
10.1.2 Finite measure spaces
Theorem 10.1.6 (Severini-Egorov-Luzin theorem for finite measure spaces). LetpΩ,F , µq be a finite measure space, let pE, dq be a metric space, and let fn : Ω Ñ E,n P N0, be strongly F/pE, dq-measurable functions. Then the following five state-ments are equivalent:
(i) It holds for all ε P p0,8q that there exists a set A P F such that µpΩzAq ă εand limnÑ8 supωPA dpfnpωq, f0pωqq “ 0 (almost uniform convergence).
(ii) It holds that there exist sets Ak P F , k P N, such that µ`
ΩzpYkPNAkq˘
“ 0,@ k P N : µpAkq ă 8, and @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 (Luzinfinite-uniform-type convergence).
(iii) It holds that there exist sets Ak P F , k P N, such that µ`
ΩzpYkPNAkq˘
“ 0and @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 (Luzin uniform-type conver-gence).
(iv) It holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 (almost sure convergence).
(v) It holds for all ε P p0,8q that limnÑ8 µ`
supmPNXrn,8q dpfm, f0q ą ε˘
“ 0.
Proof of Theorem 10.1.6. Lemma 10.1.5 ensures that piq ñ piiq. The statementthat piiq ô piiiq is clear. Lemma 10.1.4 implies that piiiq ñ pivq. Proposition 10.1.1together with the fact that the measure µ is continuous from above establishes thatpivq ñ pvq. It thus remains to prove that pvq ñ piq. For this let ε P p0,8q bearbitrary and assume that for all k P N it holds that
limnÑ8
µ`
supmPNXrn,8q dpfm, f0q ą1k
˘
“ 0. (10.9)
Observe that (10.9) shows that there exists a sequence nk P N, k P N, such that forall k P N it holds that
µ`
supmPNXrnk,8q dpfm, f0q ą1k
˘
“ µ`
Ω z
supmPNXrnk,8q dpfm, f0q ď1k
(˘
ă ε2k.
(10.10)Next let A Ď Ω be the set given by
A “ XkPN
supmPNXrnk,8q dpfm, f0q ď1k
(
“
@ k P N : @m P NX rnk,8q : dpfm, f0q ď1k
( (10.11)
and note that (10.10) shows that
µpΩzAq ďř8
k“1 µ`
Ωz
supmPNXrnk,8q dpfm, f0q ď1k
(˘
ďř8
k“1ε
2k“ ε. (10.12)
This proves that pvq ñ piq. The proof of Theorem 10.1.6 is thus completed.
10.1. EGOROV’S THEOREM ON ALMOST UNIFORM CONVERGENCE 273
10.1.3 Sigma-finite measure spaces
Corollary 10.1.7 (Severini-Egorov-Luzin theorem for sigma-finite measure spaces).Let pΩ,F , µq be a sigma-finite measure space, let pE, dq be a metric space, and letfn : Ω Ñ E, n P N0, be strongly F/pE, dq-measurable functions. Then the followingfive statements are equivalent:
(i) It holds for all ε P p0,8q, A P F that there exist a set B P pA \ Fq such thatµpAzBq ă ε and limnÑ8 supωPB dpfnpωq, f0pωqq “ 0.
(ii) It holds that there exist sets Ak P F , k P N, such that µ`
ΩzpYkPNAkq˘
“ 0,@ k P N : µpAkq ă 8, and @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 (Luzinfinite-uniform-type convergence).
(iii) It holds that there exist sets Ak P F , k P N, such that µ`
ΩzpYkPNAkq˘
“ 0and @ k P N : limnÑ8 supωPAk dpfnpωq, f0pωqq “ 0 (Luzin uniform-type conver-gence).
(iv) It holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 (almost sure convergence).
(v) It holds for all ε P p0,8q, A P F with µpAq ă 8 that
limnÑ8
µ`
AX
supmPNXrn,8q dpfm, f0q ą ε(˘
“ 0. (10.13)
Proof of Corollary 10.1.7. First, we observe that clearly piq ñ piiq. Next we notethat Lemma 10.1.4 ensures that piiq ñ piiiq. Moreover, we observe that Corl-lary 10.1.2 shows that piiiq ñ pivq. It thus remains to prove that pivq ñ piq. Wethus assume in the following that for all ε P p0,8q, A P F with µpAq ă 8 it holdsthat
limnÑ8
µ`
AX
supmPNXrn,8q dpfm, f0q ą ε(˘
“ 0. (10.14)
Moreover, we observe that the assumption that µ is sigma-finite implies that thereexists a sequence Ωk P F , k P N, such that Ω “ YkPNΩk and @ k P N : µpΩkq ă 8.Combining this with (10.14) shows that for all k P N, ε P p0,8q it holds that
limnÑ8
µ|Ωk\F`
supmPNXrn,8q dpfm|Ωk , f0|Ωkq ą ε(˘
“ 0. (10.15)
Theorem 10.1.6 hence proves that for all k P N there exist sets Ak,l P pΩk \ Fq,l P N, such that
µ|Ωk\F`
ΩkzpYlPNAk,lq˘
“ 0 and @ l P N : limnÑ8
supωPAk,l
dpfnpωq, f0pωqq “ 0.
(10.16)
274 CHAPTER 10. ADDITIONAL MATERIAL
This shows that
µpΩ z pYk,lPNAk,lqq “ µpΩ z pZkPN rYlPNAk,lsqq
“ µpZkPN rΩk z pYlPNAk,lqsq “8ÿ
k“1
µpΩk z pYlPNAk,lqq “ 0.(10.17)
Combining this and (10.16) proves that pivq ñ piq. The proof of Corollary 10.1.7 isthus completed.
10.1.4 General measure spaces
Theorem 10.1.8 (Severini-Egorov-Luzin theorem for general measure spaces). LetpΩ,F , µq be a measure space, let pE, dq be a metric space, and let fn : Ω Ñ E, n P N0,be strongly F/pE, dq-measurable functions. Then the following five statements areequivalent:
(i) It holds for all ε P p0,8q, A P F that there exist a set B P pA \ Fq such thatµpAzBq ă ε and limnÑ8 supωPB dpfnpωq, f0pωqq “ 0.
(ii) For all A P F with µpAq ă 8 it holds µ-a.s. that limnÑ8 dpfn, f0q “ 0 (almostsure convergence).
(iii) It holds for all ε P p0,8q, A P F with µpAq ă 8 that
limnÑ8
µ`
AX
supmPNXrn,8q dpfm, f0q ą ε(˘
“ 0. (10.18)
10.2 Fast convergence in probability
Let pE, dEq be a metric space and let an P E, n P N0, be a sequence of elements ofE with the property that
8ÿ
n“1
dEpan, a0q ă 8. (10.19)
Then it holds, in particular, that limnÑ8 an “ a0, i.e., it holds that panqnPN convergesto a0 in pE, dEq. The property (10.19) is sometimes referred as fast convergenceof panqnPN to a0 in pE, dEq. Fast convergence in probability ensures almost sureconvergence. This is the subject of the next lemma.
10.3. DINI’S THEOREMON POINTWISE CONVERGENCEOF CONTINUOUS FUNCTIONS275
Lemma 10.2.1 (Fast convergence implies almost sure convergence). Let pΩ,F ,Pq bea probability space, let pE, dEq be a metric space, let Xn : Ω Ñ E, n P N0, be stronglyF/pE, dEq-measurable mappings with the property that
ř8
n“1E“
min
1, dEpXn, X0q(‰
ă
8. Then it holds P-a.s. that limnÑ8Xn “ X0.
Proof of Lemma 10.2.1. First of all, observe that the Markov inequality and theassumption that
ř8
n“1E“
min
1, dEpXn, X0q(‰
ă 8 ensure that for all ε P p0, 1s itholds that
8ÿ
n“1
P`
dEpXn, X0q ě ε(˘
“
8ÿ
n“1
P`
mint1, dEpXn, X0qu ě ε(˘
ď
8ÿ
n“1
«
E“
mint1, dEpXn, X0qu‰
ε
ff
ă 8.
(10.20)
The lemma of Borel-Cantelli hence implies that for all ε P p0, 1s it holds that
Pˆ
lim supnÑ8
tdEpXn, X0q ě εu
˙
“ 0. (10.21)
This proves that for all ε P p0, 1s it holds that
PptDn P N : @m P NX rn,8q : dEpXm, X0q ă εuq “ P´
lim infnÑ8
tdEpXn, X0q ă εu¯
“ 1.
(10.22)Continuity from above of the probability measure P hence shows that
Ppt@ ε P p0,8q : Dn P N : @m P NX rn,8q : dEpXm, X0q ă εuq
“ P`
XεPp0,8q tDn P N : @m P NX rn,8q : dEpXm, X0q ă εu˘
“ limεŒ0
PptDn P N : @m P NX rn,8q : dEpXm, X0q ă εuq “ 1.(10.23)
The proof of Lemma 10.2.1 is thus completed.
10.3 Dini’s theorem on pointwise convergence of
continuous functions
See, e.g., wikipedia for the next results and their proofs.
Proposition 10.3.1 (Dini’s theorem). Let pX,X q be a compact topological space andlet fn : X Ñ r0,8q, n P N, be non-increasing continuous functions with the propertythat limnÑ8 fnpxq “ 0. Then limnÑ8 psupxPX fnpxqq “ 0.
276 CHAPTER 10. ADDITIONAL MATERIAL
Proof of Proposition 10.3.1. Throughout this proof let Xn,ε Ď X, n P N, ε P p0,8q,be the sets with the property that for all n P N, ε P p0,8q it holds that
Xn,ε “ tx P X : fnpxq ă εu “ f´1n pp´8, εqq . (10.24)
The assumption that @x P X : limnÑ8 fnpxq “ 0 implies that for all ε P p0,8q itholds that
YnPNXn,ε “ X. (10.25)
In the next step we observe that the assumption that fn : X Ñ R, n P N0, arecontinuous functions proves that for all n P N, ε P p0,8q it holds that Xn,ε P X .Combining this with (10.25) and the assumption that X is a compact set shows thatfor all ε P p0,8q there exists a natural number m P N such that
Ymn“1Xn,ε “ X. (10.26)
In addition, we observe that the asumption that @x P X, n P N : fn`1pxq ď fnpxqensures that for all ε P p0,8q, n P N it holds that Xn,ε Ď Xn`1,ε. This and (10.26)prove that for all ε P p0,8q there exists a natural number m P N such that
X “ Ymn“1Xn,ε “ Xm,ε “ X
8n“mXn,ε. (10.27)
The proof of Proposition 10.3.1 is thus completed.
Theorem 10.3.2 (Dini’s theorem). Let pX,X q be a compact topological space, letpE, dq be a metric space, and let fn : X Ñ E, n P N0, be continuous functionswith the property that @x P X, n P N : d
`
f0pxq, fn`1pxq˘
ď d`
f0pxq, fnpxq˘
andlimmÑ8 fmpxq “ f0pxq. Then
limnÑ8
“
supxPX d`
f0pxq, fnpxq˘‰
“ 0. (10.28)
Proof of Theorem 10.3.2. Throughout this proof let gn : X Ñ r0,8q, n P N, be thefunctions with the property that for all n P N, x P X it holds that
gnpxq “ d`
fnpxq, f0pxq˘
. (10.29)
Proposition 10.3.1 then proves that limnÑ8
`
supxPX gnpxq˘
“ 0. The proof of Theo-rem 10.3.2 is thus completed.
10.3. DINI’S THEOREMON POINTWISE CONVERGENCEOF CONTINUOUS FUNCTIONS277
10.3.1 On the compactness of the argument space
The next example shows that the assumption in Theorem 10.3.2 that X is a compactset can not be avoided.
Example 10.3.3. Let fn : r0,8q Ñ R, n P N0, be the functions with the propertythat for all n P N, x P r0, 1s it holds that fnpxq “
xn
and f0pxq “ 0. Then
• it holds for all n P N0 that fn P Cpr0,8q,Rq,
• it holds for all x P r0,8q, n P N that
|fn`1pxq ´ f0pxq| “x
pn` 1qďx
n“ |fnpxq ´ f0pxq| , (10.30)
• it holds for all x P r0,8q that limnÑ8 fnpxq “ f0pxq, and
• it holds that
limnÑ8
«
supxPr0,8q
|fnpxq ´ f0pxq|
ff
“ limnÑ8
«
supxPr0,8q
xn
ff
“ 8. (10.31)
10.3.2 On the monotonicity of the approximating functions
The next example shows that the assumption in Theorem 10.3.2 that it holds forevery x P X that the sequence dpf0pxq, fnpxqq, n P N, is non-increasing can not be
278 CHAPTER 10. ADDITIONAL MATERIAL
avoided.
Example 10.3.4. Let gr,ε : R Ñ r0, 1s, r P R, ε P p0,8q, be the functions with theproperty that for all r P R, ε P p0,8q it holds that
gr,εpxq “
$
’
’
&
’
’
%
0 : x P Rzpr ´ ε, r ` εq
1εpx´ pr ´ εqq : x P rr ´ ε, rs
1εppr ` εq ´ xq : x P rr, r ` εs
(10.32)
and let fn : r0, 1s Ñ R, n P N0, be the functions with the property that for all n P N,x P r0, 1s it holds that fnpxq “ gn´1,2´npxq and f0pxq “ 0. Then
• it holds for all n P N0 that fn P Cpr0, 1s,Rq,
• it holds for all x P r0, 1s that limnÑ8 fnpxq “ f0pxq, and
• it holds that
limnÑ8
«
supxPr0,1s
|fnpxq ´ f0pxq|
ff
“ limnÑ8
«
supxPr0,1s
gn´1,2´npxq
ff
“ 1. (10.33)
10.3.3 On the continuity of the approximating functions
The next example shows that the assumption in Theorem 10.3.2 that the functionsfn, n P N, are continuous can not be avoided.
Example 10.3.5. Let fn : r0, 1s Ñ R, n P N0, be the functions with the propertythat for all n P N, x P r0, 1s it holds that fnpxq “ 1
R
p0,2´nqpxq and f0pxq “ 0. Then
• it holds that f0 P Cpr0, 1s,Rq,
• it holds for all x P r0, 1s, n P N that
|fn`1pxq ´ f0pxq| “ 1R
p0,2´pn`1qqpxq ď 1Rp0,2´nqpxq “ |fnpxq ´ f0pxq| , (10.34)
• it holds for all x P r0, 1s that limnÑ8 fnpxq “ f0pxq, and
• it holds that
limnÑ8
«
supxPr0,1s
|fnpxq ´ f0pxq|
ff
“ limnÑ8
«
supxPr0,1s
1p0,2´nqpxq
ff
“ 1. (10.35)
10.3. DINI’S THEOREMON POINTWISE CONVERGENCEOF CONTINUOUS FUNCTIONS279
10.3.4 On the continuity of the limit function
The next example shows that the assumption in Theorem 10.3.2 that the functionf0 is continuous can not be avoided.
Example 10.3.6. Let fn : r0, 1s Ñ R, n P N0, be the functions with the propertythat for all n P N, x P r0, 1s it holds that fnpxq “ xn and f0pxq “ 1
R
t1upxq. Then
• it holds for all n P N that fn P Cpr0, 1s,Rq,
• it holds for all x P r0, 1s, n P N that
|fn`1pxq ´ f0pxq| “ 1R
r0,1qpxq ¨ xpn`1q
ď 1Rr0,1qpxq ¨ xn“ |fnpxq ´ f0pxq| , (10.36)
• it holds for all x P r0, 1s that limnÑ8 fnpxq “ f0pxq, and
• it holds that
limnÑ8
«
supxPr0,1s
|fnpxq ´ f0pxq|
ff
“ limnÑ8
«
supxPr0,1q
xn
ff
“ 1. (10.37)
280 CHAPTER 10. ADDITIONAL MATERIAL
Chapter 11
Solutions to selected exercises
11.1 Chapter 2
11.1.1 Solution to Exercise 2.2.6
Lemma 11.1.1. Let pΩ,Fq be a measurable space, let pE, dEq be a metric space andlet f : Ω Ñ E be a function. Then f is F/BpEq-measurable if and only if it holdsfor all ϕ P CpE,Rq that ϕ ˝ f is F/BpRq-measurable.
Proof of Lemma 11.1.1. First of all, recall that every every ϕ P CpE,Rq is BpEq/BpRq-measurable. This shows that if f is F/BpEq-measurable, then it holds for everyϕ P CpE,Rq that the composition ϕ ˝ f is F/BpRq-measurable. It thus remainsto prove that if for every ϕ P CpE,Rq it holds that ϕ ˝ f is F/BpRq-measurable,then f is F/BpEq-measurable. This, in turn, is an immediate consequence fromProposition 2.2.5. The proof of Lemma 11.1.1 is thus completed.
11.1.2 Solution to Exercise 2.4.4
Lemma 11.1.2 (Example of a measurable but not strongly measurable function).It holds that the function IdL2p#R;|¨|
Rq is B
`
L2p#R; |¨|Rq˘
/B`
L2p#R; |¨|Rq˘
-measurable
but not B`
L2p#R; |¨|Rq˘
/`
L2p#R; |¨|Rq, ¨L2p#R;|¨|
Rq
˘
-measurable.
Proof of Lemma 11.1.2. Clearly, IdL2p#R;|¨|Rq is continuous. Hence, IdL2p#R;|¨|
Rq is
also B`
L2p#R; |¨|Rq˘
/B`
L2p#R; |¨|Rq˘
-measurable. Next note that Example 3.3.8shows that L2p#R; |¨|
Rq is not separable. Combining this with the fact that
im`
IdL2p#R;|¨|Rq
˘
“ L2p#R; |¨|
Rq (11.1)
281
282 CHAPTER 11. SOLUTIONS TO SELECTED EXERCISES
completes the proof of Lemma 11.1.2.
11.2 Chapter 3
11.2.1 Solution to Exercise 3.5.2
Lemma 11.2.1. Let K P tR,Cu, let pH, 〈¨, ¨〉H , ¨Hq be a K-Hilbert space, and let
A : DpAq Ď H Ñ H be a diagonal linear operator. Then DpAqH“ H.
Proof of Lemma 11.2.1. The assumption that A : DpAq Ď H Ñ H is a diagonallinear operator ensures that there exists an orthonormal basis B Ď H of H and afunction λ : BÑ K such that
DpAq “
#
v P H :ÿ
bPB
|λb|2|〈b, v〉H |
2ă 8
+
(11.2)
and such that for all v P DpAq it holds that
Av “ÿ
bPB
λb 〈b, v〉H b. (11.3)
Equation (11.2) implies thatB Ď DpAq. (11.4)
Moreover, the fact that B is an orthonormal basis ensures that BH“ H. This and
(11.4) prove that DpAqH“ H. The proof of Lemma 11.2.1 is thus completed.
11.3. CHAPTER 7 283
11.3 Chapter 7
11.3.1 Solution to Exercise 7.1.15
Proposition 11.3.1. Assume the setting in Subsection 7.1.2 and let X : r0, T sˆΩ ÑHγ be the up to modifications unique pFtqtPr0,T s-predictable stochastic process whichsatisfies suptPr0,T s XtLppP;¨Hγ q
ă 8 and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (11.5)
Then for all r P“
γ,mint1 ` γ ´ η, 12 ` βu˘
, ε P`
0,mint1 ` γ ´ η, 12 ` βu ´ r˘
itholds that
supt1,t2Pr0,T s,t1‰t2
¨
˝
›
›
`
Xt1 ´ et1AX0
˘
´`
Xt2 ´ et2AX0
˘›
›
LppP;¨Hr q
|t1 ´ t2|ε
˛
‚
ď
«
supsPr0,T s
F pXsqLppP;¨Hγ´ηq
ff
2T p1`γ´η´r´εq
p1` γ ´ η ´ ε´ rq
`
«
supsPr0,T s
BpXsqLppP;¨HSpU,Hβq
ff
a
p pp´ 1qT p12`β´r´εq
p1` 2β ´ 2r ´ 2εq12ă 8.
(11.6)
Proof of Proposition 11.3.1. First of all, note that Theorem 4.7.6 and Lemma 4.7.7imply that for all r P rγ, 1` γ ´ ηq, ε P p0, 1` γ ´ η ´ rq, t1, t2 P r0, T s with t1 ă t2
284 CHAPTER 11. SOLUTIONS TO SELECTED EXERCISES
it holds that›
›
›
›
ż t1
0
ept1´sqAF pXsq ds´
ż t2
0
ept2´sqAF pXsq ds
›
›
›
›
LppP;¨Hr q
ď
ż t2
t1
›
›ept2´sqAF pXsq›
›
LppP;Hrqds`
ż t1
0
›
›
`
ept1´sqA ´ ept2´sqA˘
F pXsq›
›
LppP;¨Hr qds
ď
«
supsPr0,T s
F pXsqLppP;¨Hγ´ηq
ff
¨
„ż t2
t1
pt2 ´ sqγ´η´r ds`
ż t1
0
pt1 ´ sqγ´η´r´ε
pt2 ´ t1qε ds
“
«
supsPr0,T s
F pXsqLppP;¨Hγ´ηq
ff«
pt2 ´ t1qp1`γ´η´rq
p1` γ ´ η ´ rq`pt2 ´ t1q
εpt1q
p1`γ´η´r´εq
p1` γ ´ η ´ r ´ εq
ff
ď
«
supsPr0,T s
F pXsqLppP;¨Hγ´ηq
ff
„
2T p1`γ´η´r´εq pt2 ´ t1qε
p1` γ ´ η ´ ε´ rq
.
(11.7)
Moreover, observe that Theorem 4.7.6 and Lemma 4.7.7 ensure that for all r Prγ, 12` βq, ε P p0, 12` β ´ rq, t1, t2 P r0, T s with t1 ă t2 it holds that
›
›
›
›
ż t1
0
ept1´sqABpXsq dWs ´
ż t2
0
ept2´sqABpXsq dWs
›
›
›
›
2
LppP;¨Hr q
ďp pp´1q
2
ż t2
t1
›
›ept2´sqABpXsq›
›
2
LppP;¨HSpU,Hrqqds
`p pp´1q
2
ż t1
0
›
›
`
ept1´sqA ´ ept2´sqA˘
BpXsq›
›
2
LppP;¨HSpU,Hrqqds
ďp pp´1q
2
«
supsPr0,T s
BpXsqLppP;¨HSpU,Hβq
ff2
¨
„ż t2
t1
pt2 ´ sqp2β´2rq ds`
ż t1
0
pt1 ´ sqp2β´2r´2εq
pt2 ´ t1q2ε ds
ďp pp´1q
2
«
supsPr0,T s
BpXsqLppP;¨HSpU,Hβq
ff22 pt2 ´ t1q
2εpt2q
p1`2β´2r´2εq
p1` 2β ´ 2r ´ 2εq.
(11.8)
Combining (11.7) and (11.8) completes the proof of Proposition 11.3.1.
11.3. CHAPTER 7 285
11.3.2 Solution to Exercise 7.1.16
Corollary 11.3.2. Assume the setting in Subsection 7.1.2, let δ P rγ,8q, assumethat ξ P LppP; ¨Hδq, and let X : r0, T s ˆ Ω Ñ Hγ be the up to modifications uniquepFtqtPr0,T s-predictable stochastic process which satisfies suptPr0,T s XtLppP;¨Hγ q
ă 8
and which is a mild solution of the SPDE
dXt “ rAXt ` F pXtqs dt`BpXtq dWt, t P r0, T s, X0 “ ξ. (11.9)
Then for all r P“
γ,mint1 ` γ ´ η, 12 ` βu˘
, ε P`
0,mint1 ` γ ´ η, 12 ` βu ´ r˘
itholds that
supt1,t2Pr0,T s,t1‰t2
¨
˝
|mintt1, t2u|maxtr`ε´δ,0u
Xt1 ´Xt2LppP;¨Hr q
|t1 ´ t2|ε
˛
‚
ď X0LppP;¨Hmintδ,r`εuq`
«
supsPr0,T s
F pXsqLppP;¨Hγ´ηq
ff
2T p1`γ´η´mintδ,r`εuq
p1` γ ´ η ´ ε´ rq
`
«
supsPr0,T s
BpXsqLppP;¨HSpU,Hβq
ff ?2T p12`β´mintδ,r`εuq
p1` 2β ´ 2r ´ 2εq12ă 8.
(11.10)
Proof of Corollary 11.3.2. Note that Lemma 4.7.7 ensures that for all r P“
γ,mint1`γ ´ η, 12` βu
˘
, ε P`
0,mint1` γ ´ η, 12` βu ´ r˘
it holds that
supt1,t2Pr0,T s,t1‰t2
¨
˝
|mintt1, t2u|maxtr`ε´δ,0u
›
›et1AX0 ´ et2AX0
›
›
LppP;¨Hr q
|t1 ´ t2|ε
˛
‚
ď supt1,t2Pr0,T s,t1‰t2
˜
|mintt1,t2u|maxtr`ε´δ,0u
p´Aqr´mintδ,r`εupet1A´et2AqLpHq X0LppP;¨Hmintδ,r`εuq
|t1´t2|ε
¸
ď supt1,t2Pr0,T s,t1ăt2
ˆ
|t1|maxtr`ε´δ,0u
p´Aqr`ε´mintδ,r`εu et1ALpHq X0LppP;¨Hmintδ,r`εuq
˙
“ supt1,t2Pr0,T s,t1ăt2
ˆ
|t1|maxtr`ε´δ,0u
p´Aqmaxtr`ε´δ,0u et1ALpHq X0LppP;¨Hmintδ,r`εuq
˙
ď X0LppP;¨Hmintδ,r`εuq.
(11.11)
286 CHAPTER 11. SOLUTIONS TO SELECTED EXERCISES
Combining this with the triangle inequality and Proposition 11.3.1 completes theproof of Corollary 11.3.2.
11.3.3 Solution to Exercise 7.2.2
Lemma 11.3.3 (Variances). Let T P p0,8q, let pΩ,F ,P, pFtqtPr0,T sq be a stochasticbasis, let pH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|q, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rq
, ¨L2pBorelp0,1q;|¨|Rqq,
let pWtqtPr0,T s be a cylindrical IdH-Wiener process w.r.t. pFtqtPr0,T s, and let X : r0, T sˆΩ Ñ H be a mild solution process of the SPDE
dXtpxq “B2
Bx2Xtpxq dt` dWtpxq, Xtp0q “ Xtp1q “ 0, X0pxq “ 0 (11.12)
for x P p0, 1q, t P r0, T s. Then it holds thatş1
0E“
|XT pxq|2‰
dx “ř8
n“11´e´2π2n2T
2π2n2 ‰ř8
n“11´e´2π2n2T
π2n2 .
Proof of Lemma 11.3.3. Throughout this proof let A : DpAq Ď H Ñ H be the Lapla-cian with Dirichlet boundary conditions on H. Then observe that for all t P r0, T s itholds that
E
«
›
›
›
›
ż t
0
ept´sqA dWs
›
›
›
›
2
H
ff
“
ż t
0
›
›ept´sqA›
›
2
HSpHqds “
ż t
0
›
›esA›
›
2
HSpHqds
“
8ÿ
n“1
ż t
0
e´2π2n2s ds “8ÿ
n“1
˜
1´ e´2π2n2t
2π2n2
¸
.
(11.13)
The proof of Lemma 11.3.3 is thus completed.
11.4 Chapter 8
11.4.1 Solution to Exercise 8.1.6
Lemma 11.4.1. Let r P p1,8q. Thenř8
n“11
n |lnpn`1q|ră 8.
11.4. CHAPTER 8 287
Proof of Lemma 11.4.1. Observe that
8ÿ
n“1
1
n |lnpn` 1q|r“
1
lnp2q`
1
2 lnp3q`
8ÿ
n“3
1
n |lnpn` 1q|r
ď1
lnp2q`
1
2 lnp3q`
8ÿ
n“3
1
n |lnpnq|rď
1
lnp2q`
1
2 lnp3q`
8ÿ
n“3
ż n
n´1
1
n |lnpnq|rdx
ď1
lnp2q`
1
2 lnp3q`
8ÿ
n“3
ż n
n´1
1
x |lnpxq|rdx
“1
lnp2q`
1
2 lnp3q`
ż 8
2
1
x |lnpxq|rdx “
1
lnp2q`
1
2 lnp3q`
ż 8
lnp2q
1
xrdx
“1
lnp2q`
1
2 lnp3q`
1
pr ´ 1q |lnp2q|pr´1qă 8.
(11.14)
The proof of Lemma 11.4.1 is thus completed.
Proposition 11.4.2 (Lower bounds on the convergence speed). Let A : DpAq ĎL2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be the Laplace operator with Dirichlet bound-ary conditions on L2pBorelp0,1q; |¨|Rq, let pHr, 〈¨, ¨〉Hr , ¨Hrq, r P R, be a family of in-terpolation spaces associated to ´A, let r P R, ρ P r0,8q, let en P L
2pBorelp0,1q; |¨|Rq,n P N, satisfy that for all n P N and Borelp0,1q-almost all x P p0, 1q it holds that
enpxq “?
2 sinpnπxq, and let v P Hr satisfy
v “8ÿ
k“1
k´12´2r´2ρ
|lnpk ` 1q|´1Rek. (11.15)
Then
(i) it holds that v P Hr`ρzpYεPp0,8qHr`ρ`εq,
(ii) it holds for all ε P p´8, 0s that
supnPN
˜
np2ρ`εq
›
›
›
›
›
v ´nÿ
k“1
〈ek, v〉H ek
›
›
›
›
›
Hr
¸
ă 8, (11.16)
and
288 CHAPTER 11. SOLUTIONS TO SELECTED EXERCISES
(iii) it holds for all ε P p0,8q that
supnPN
˜
np2ρ`εq
›
›
›
›
›
v ´nÿ
k“1
〈ek, v〉H ek
›
›
›
›
›
Hr
¸
“ 8. (11.17)
Proof of Proposition 11.4.2. Observe that Lemma 11.4.1 ensures that
8ÿ
k“1
›
›k´12´2r´2ρ
|lnpk ` 1q|´1Rek›
›
2
Hr`ρ
“
8ÿ
k“1
k´1´4r´4ρ|lnpk ` 1q|´2
Rek
2Hr`ρ
“ π4r`4ρ8ÿ
k“1
k´1|lnpk ` 1q|´2
Ră 8.
(11.18)
This implies that v P Hr`ρ. Example 8.1.5 therefore proves Item (ii). In addition,observe that for all ε P p0,8q it holds that
supnPN
¨
˝np4ρ`2εq
›
›
›
›
›
v ´nÿ
k“1
〈ek, v〉H ek
›
›
›
›
›
2
Hr
˛
‚
“ supnPN
˜
np4ρ`2εq8ÿ
k“n`1
〈ek, v〉H ek2Hr
¸
“ supnPN
˜
np4ρ`2εq8ÿ
k“n`1
|〈ek, v〉H |2Rπ4r k4r
¸
“ supnPN
˜
np4ρ`2εq8ÿ
k“n`1
k´1´4r´4ρ|lnpk ` 1q|´2
Rπ4r k4r
¸
“ π4r supnPN
˜
np4ρ`2εq8ÿ
k“n`1
k´1´4ρ|lnpk ` 1q|´2
R
¸
“ 8.
(11.19)
This proves Item (iii). Example 8.1.5 and (11.19) imply that v R pYεPp0,8qHr`ρ`εq.This together with the fact that v P Hr`ρ proves Item (i). The proof of Proposi-tion 11.4.2 is thus completed.
11.4.2 Solution to Exercise 8.1.15
11.4. CHAPTER 8 289
1 function E x e r c i s e P l o t S p a t i a l ( )2 Ngrid = 2 . ˆ ( 0 : 1 9 ) ; Errs = Ngrid ;3 t ic4 for n = 1 : length ( Ngrid )5 Errs (n) = MCErr( Ngrid (n ) , 2ˆ23 , 2ˆ5 ) ;6 end7 toc8 loglog ( Ngrid , Errs ) ;9 hold on
10 loglog ( Ngrid , Ngrid .ˆ( ´1)∗0 .3 , ’ :∗ r ’ ) ;11 loglog ( Ngrid , Ngrid . ˆ ( ´0 .5 )∗0 .3 , ’ : or ’ ) ;12 loglog ( Ngrid , Ngrid . ˆ ( ´0 .25 )∗0 .3 , ’ :∗ r ’ ) ;13 t i t l e ( ’ Error Galerk in approximation ’ ) ;14 xlabel ( ’ S p a t i a l dimension ’ ) ;15 ylabel ( ’ Root mean square e r r o r ’ ) ;16 end1718 function error = SqrdSampleErr (N, top )19 A = ´pi ˆ2∗ ( (N+1): top ) . ˆ 2 ;20 var = (exp(2∗A)´1)./A/2 ;21 gsq = randn (1 , top N) . ˆ 2 ;22 error = sum( var .∗ gsq ) ;23 end2425 function error = MCErr(N, top ,M)26 error = 0 ;27 for m=1:M28 error = error + SqrdSampleErr (N, top ) ;29 end30 error = sqrt ( error/M) ;31 end
Matlab code 11.1: A Matlab code for a solution of Exercise 8.1.15.
290 CHAPTER 11. SOLUTIONS TO SELECTED EXERCISES
100
101
102
103
104
105
106
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Error Galerkin approximation
Spatial dimension
Ro
ot
me
an
sq
ua
re e
rro
r
Figure 11.1: Result of a call of the Matlab code 11.1.
11.4. CHAPTER 8 291
11.4.3 Solution to Exercise 8.2.5
Lemma 11.4.3 (Elementary estimates for the zeta function). Let s P p1,8q. Then
8ÿ
n“1
n´s ďs
ps´ 1q. (11.20)
Proof of Lemma 11.4.3. Observe that
8ÿ
n“1
n´s “ 1`8ÿ
n“2
ż n
n´1
n´s du ď 1`8ÿ
n“2
ż n
n´1
u´s du
ď 1`
ż 8
1
u´s du “ 1`1
ps´ 1q“
s
ps´ 1q.
(11.21)
The proof of Lemma 11.4.3 is thus completed.
Proposition 11.4.4. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq bethe Laplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, letT P p0,8q, N P N, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic basis, let pWtqtPr0,T s be acylindrical IdH-Wiener process w.r.t. pFtqtPr0,T s, let Y : r0, T sˆΩ Ñ H be a naturally-interpolated exponential Euler approximation for the SPDE
dXt “ AXt dt` dWt, t P r0, T s, X0 “ 0 (11.22)
with time step size TN, and let X : r0, T s ˆ Ω Ñ H be an pFtqtPr0,T s-predictablestochastic process which fulfills that for all t P r0, T s it holds P-a.s. that
Xt “
ż t
0
eApt´sq dWs. (11.23)
Then it holds for all r P r0, 14q that`
E“
XT ´ YT 2H
‰˘12ď T r
Nr p12´2rq12ď T r
Nr p1´4rq.
Proof of Proposition 11.4.4. Observe that for all r P r0,8q it holds that
E“
XT ´ YT 2H
‰
“ E
«
›
›
›
›
ż T
0
`
epT´sqA ´ epT´tsuT N qA˘
dWs
›
›
›
›
2
H
ff
“
ż T
0
›
›epT´sqA ´ epT´tsuT N qA›
›
2
HSpHqds
“
ż T
0
›
›p´Aqr epT´sqA p´Aq´r`
eps´tsuT N qA ´ IdH˘›
›
2
HSpHqds
ď
ż T
0
›
›p´Aqr epT´sqA›
›
2
HSpHq
›
›p´Aq´r`
eps´tsuT N qA ´ IdH˘›
›
2
LpHqds.
(11.24)
292 CHAPTER 11. SOLUTIONS TO SELECTED EXERCISES
Lemma 4.7.7 hence proves that for all r P r0,8q it holds that
E“
XT ´ YT 2H
‰
ď“
TN
‰2rż T
0
›
›p´Aqr epT´sqA›
›
2
HSpHqds “
“
TN
‰2r8ÿ
n“1
ż T
0
n4r π4r e´2π2n2s ds
““
TN
‰2r8ÿ
n“1
1´ e´2π2n2T
2πp2´4rqnp2´4rqď“
TN
‰2r8ÿ
n“1
1
2πp2´4rq np2´4rq
“ πp4r´2q
2
“
TN
‰2r
«
8ÿ
n“1
np4r´2q
ff
.
(11.25)
Lemma 11.4.3 therefore implies that for all r P r0, 14q it holds that
E“
XT ´ YT 2H
‰
ď πp4r´2q
2
“
TN
‰2r p2´ 4rq
p1´ 4rqď
T 2r
p12´ 2rqN2r. (11.26)
The proof of Proposition 11.4.4 is thus completed.
11.5 Chapter 9
11.5.1 Solution to Exercise 9.4.1
Proposition 11.5.1. Let A : DpAq Ď L2pBorelp0,1q; |¨|Rq Ñ L2pBorelp0,1q; |¨|Rq be theLaplace operator with Dirichlet boundary conditions on L2pBorelp0,1q; |¨|Rq, let T P
p0,8q, N P N, ϕ : L2pBorelp0,1q; |¨|Rq Ñ R satisfy that for all v P L2pBorelp0,1q; |¨|Rq
it holds that ϕpvq “ v2L2pBorelp0,1q;|¨|Rq, let pΩ,F ,P, pFtqtPr0,T sq be a stochastic ba-
sis, let pWtqtPr0,T s be a cylindrical IdL2pBorelp0,1q;|¨|Rq-Wiener process w.r.t. pFtqtPr0,T s,
let Y : r0, T s ˆ Ω Ñ L2pBorelp0,1q; |¨|Rq be a naturally-interpolated exponential Eulerapproximation for the SPDE
dXt “ AXt dt` dWt, t P r0, T s, X0 “ 0 (11.27)
with time step size TN, and let X : r0, T s ˆ Ω Ñ L2pBorelp0,1q; |¨|Rq be an pFtqtPr0,T s-predictable stochastic process which fulfills that for all t P r0, T s it holds P-a.s. that
Xt “
ż t
0
eApt´sq dWs. (11.28)
Then it holds for all r P“
0, 12˘
thatˇ
ˇE“
ϕpXT q‰
´ E“
ϕpYT q‰ˇ
ˇ ď T r
Nr π?
2 p1´2rqď
T r
Nr p12´rq.
11.5. CHAPTER 9 293
Proof of Proposition 11.5.1. Throughout this proof let λn P R, n P N, be the realnumbers with the property that for all n P N it holds that λn “ π2n2 and letpH, 〈¨, ¨〉H , ¨Hq be the R-Hilbert space such that
pH, 〈¨, ¨〉H , ¨Hq “ pL2pBorelp0,1q; |¨|Rq, ¨L2pBorelp0,1q;|¨|Rq
, 〈¨, ¨〉L2pBorelp0,1q;|¨|Rqq.
(11.29)Then observe that
ErϕpXT qs “
ż t
0
›
›ept´sqA›
›
2
HSpHqds “
8ÿ
n“1
ż t
0
e´2λnpt´sq ds (11.30)
and
ErϕpYT qs “
ż t
0
›
›ept´tsuT N qA›
›
2
HSpHqds “
8ÿ
n“1
ż t
0
e´2λnpt´tsuT N q ds. (11.31)
Combining (11.30) and (11.31) proves that for all r P“
0, 12˘
it holds that
|ErϕpXT qs ´ErϕpYT qs| “8ÿ
n“1
ż t
0
e´2λnpt´sq`
1´ e´2λnps´tsuT N q˘
ds
ď
8ÿ
n“1
ż t
0
e´2λnpt´sq`
1´ e´2λnps´tsuT N q˘rds
ď
8ÿ
n“1
ż t
0
e´2λnpt´sq`
2λnps´ tsuT Nq˘rds ď
8ÿ
n“1
2rpλnqrT r
N r
ż t
0
e´2λns ds
ď
8ÿ
n“1
T r
2p1´rq pλnqp1´rqN r
“T r πp2r´2q
2p1´rqN r
«
8ÿ
n“1
n´p2´2rq
ff
ďT r πp2r´2q
2p1´rqN r¨p2´ 2rq
p1´ 2rqď
T r
N r p12´ rq
«
supsPr0,12q
p1´ sq
p2π2qp1´sq
ff
ďT r
N r p12´ rq¨p12q
p2π2q12“
T r
π?
2 p1´ 2rqN r.
(11.32)
The proof of Proposition 11.5.1 is thus completed.
294 CHAPTER 11. SOLUTIONS TO SELECTED EXERCISES
Bibliography
[1] Brzezniak, Z., van Neerven, J. M. A. M., Veraar, M. C., and Weis,L. Ito’s formula in UMD Banach spaces and regularity of solutions of the Zakaiequation. J. Differential Equations 245, 1 (2008), 30–58.
[2] Cohn, D. L. Measure theory. Birkhauser Boston Inc., Boston, MA, 1993.Reprint of the 1980 original.
[3] Conus, D. The Non-linear Stochastic Wave Equation in High Dimensions:Existence, Holder-continuity and Ito-Taylor Expansion. Ecole PolytechniqueFedederale De Lausanne, 2008. Dissertation.
[4] Conus, D., and Dalang, R. C. The non-linear stochastic wave equation inhigh dimensions. electronic journal of probability. Electronic Journal of Proba-bility 13 (2008), 629–670.
[5] Conus, D., Jentzen, A., and Kurniawan, R. Weak convergence rates ofspectral Galerkin approximations for stochastic evolution equations with non-linear diffusion coefficients. working paper (2014), XX pages.
[6] Da Prato, G., Jentzen, A., and Rockner, M. A mild Ito formula forSPDEs. arXiv:1009.3526 (2012), 1–39.
[7] Da Prato, G., and Zabczyk, J. Stochastic equations in infinite dimen-sions, vol. 44 of Encyclopedia of Mathematics and its Applications. CambridgeUniversity Press, Cambridge, 1992.
[8] Debussche, A., and Printems, J. Weak order for the discretization of thestochastic heat equation. Math. Comp. 78, 266 (2009), 845–863.
[9] Hardy, G. H., Littlewood, J. E., and Polya, G. Inequalities. CambridgeMathematical Library. Cambridge University Press, Cambridge, 1988. Reprintof the 1952 edition.
295
296 BIBLIOGRAPHY
[10] Henry, D. Geometric theory of semilinear parabolic equations, vol. 840 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1981. 348 pages.
[11] Heuser, H. Lehrbuch der Analysis. Teil 2, sixth ed. Mathematische Leitfaden.[Mathematical Textbooks]. B. G. Teubner, Stuttgart, 1991.
[12] Hutzenthaler, M., Jentzen, A., and Kloeden, P. E. Divergence ofthe multilevel Monte Carlo Euler method for nonlinear stochastic differentialequations. To appear in Ann. Appl. Probab. (2013); arXiv:1105.0226 (2011),31 pages.
[13] Hutzenthaler, M., Jentzen, A., and Kloeden, P. E. Strong and weakdivergence in finite time of Euler’s method for stochastic differential equationswith non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. AMath. Phys. Eng. Sci. 467 (2011), 1563–1576.
[14] Jentzen, A., and Kloeden, P. Taylor Approximations for Stochastic PartialDifferential Equations, vol. 83 of CBMS-NSF Regional Conference Series inApplied Mathematics. Society for Industrial and Applied Mathematics (SIAM),Philadelphia, PA, 2012.
[15] Jentzen, A., and Kurniawan, R. Weak convergence rates for Euler-typeapproximations of semilinear stochastic evolution equations with nonlinear dif-fusion coefficients. arXiv:1501.03539 (2015), 1–51.
[16] Karatzas, I., and Shreve, S. E. Brownian motion and stochastic calculus,second ed., vol. 113 of Graduate Texts in Mathematics. Springer-Verlag, NewYork, 1991.
[17] Klenke, A. Probability theory. Universitext. Springer-Verlag London Ltd.,London, 2008. A comprehensive course, Translated from the 2006 German orig-inal.
[18] Klenke, A. Probability theory. Universitext. Springer-Verlag London Ltd.,London, 2008. A comprehensive course, Translated from the 2006 German orig-inal.
[19] Kovacs, M., Larsson, S., and Lindgren, F. Weak convergence of finiteelement approximations of linear stochastic evolution equations with additivenoise. BIT Numerical Mathematics (2011), 24 pages.
[20] Kuhn, C. Stochastische Analysis mit Finanzmathematik. 2004.
BIBLIOGRAPHY 297
[21] Kurniawan, R. Numerical approximations of stochastic partial differentialequations with non-globally Lipschitz continuous nonlinearities. University ofZurich and ETH Zurich, Zurich, Switzerland, 2014. 74 pages. Master thesis.
[22] Lindner, F., and Schilling, R. L. Weak order for the discretization of thestochastic heat equation driven by impulsive noise. arXiv:0911.4681v2 (2010),29 pages.
[23] Pazy, A. Semigroups of linear operators and applications to partial differentialequations, vol. 44 of Applied Mathematical Sciences. Springer-Verlag, New York,1983.
[24] Prevot, C., and Rockner, M. A concise course on stochastic partial dif-ferential equations, vol. 1905 of Lecture Notes in Mathematics. Springer, Berlin,2007. 144 pages.
[25] Reed, M., and Simon, B. Methods of modern mathematical physics. I. Func-tional analysis. Academic Press, New York, 1972.
[26] Stein, E. M. Singular integrals and differentiability properties of functions.Princeton Mathematical Series, No. 30. Princeton University Press, Princeton,N.J., 1970.
[27] Van Neerven, J. Stochastic Evolution Equations. Lec-ture notes (2007), 234 pages. Available online athttp://fa.its.tudelft.nl/„neerven/publications/papers/ISEM.pdf.
[28] van Neerven, J. M. A. M., Veraar, M. C., and Weis, L. Stochasticevolution equations in UMD Banach spaces. J. Funct. Anal. 255, 4 (2008),940–993.
[29] Werner, D. Funktionalanalysis, extended ed. Springer-Verlag, Berlin, 2005.