glauber dynamics in continuum - uni-bielefeld.de

Glauber Dynamics in Continuum

Yuri Kondratievjoint work with O.Kutovyi and R.Minlos

Bielefeld, Germany

October 28, 2008

Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 1 / 24

Setup General facts and notations

The configuration space:

Γ :={γ ⊂ Rd

∣∣ |γ ∩ Λ| <∞ for all compact Λ ⊂ Rd}.

| · | - cardinality of the set.

Remark: Γ is a Polish space.

n-point configuration space:

Γ(n) :={η ⊂ Rd | |η| = n

}, n ∈ N0.

The space of finite configurations:

Γ0 :=⊔n∈N0

Γ(n).

Glauber dynamics with competition

Potential

φ : Rd 7→ R ∪ {+∞}:

(S) (Stability) There exists B > 0 such that, for any η ∈ Γ0, |η| ≥ 2.

E(η) :=∑

{x, y}⊂η

φ(x− y) ≥ −B|η|.

(SI) (Strong Integrability) For any β > 0,

Cst(β) :=∫

|1− exp [βφ(x)]|dx <∞.

Relative energy: γ ∈ Γ, x ∈ Rd \ γ

E(x, γ) :={ ∑

y∈γ φ(x− y), if∑y∈γ |φ(x− y)| <∞,

+∞, otherwise.

Gibbs measures

∑x∈γ

F (γ, x)µ(dγ) =∫

e−βE(x, γ)F (γ ∪ x, x)zdxµ(dγ)

for any measurable function F : Γ× Rd → [0,+∞].

G(φ, z, β) - the set of all Gibbs measures

Remark: If φ is stable and

C(β) :=∫

|1− e−βφ(x)|dx < z−1e−1−2βB

then the class G(φ, z, β) is non-empty. Moreover,

kµ(η) ≤ constC(β)−|η|, η ∈ Γ0.

Dirichlet forms

E+(F, F ) =∫

|F (γ ∪ x)− F (γ)|2 exp(−E(x, γ))zdxdµ(γ)

G+-dynamics

E−(F, F ) =∫

|F (γ ∪ x)− F (γ)|2zdxdµ(γ)

G−-dynamics

Corresponding equilibrium dynamics:K/Lytvynov, K/Lytvynov/Rockner

Spectral properties of equilibrium Markov generators:Bertini/Cancrini/Cesi, K/Lytvynov, Wu, K/Minlos/Zhizhina

We will study G− Glauber non-equilibrium stochastic dynamics

Pre-generator

Markov pre-generator on Γ:

(LF )(γ) :=∑x∈γ

eβE(x, γ)D−x F (γ) + z

D+x F (γ)dx,

D−x F (γ) = F (γ \ x)− F (γ) death of x ∈ γ;

D+x F (γ) = F (γ ∪ x)− F (γ) birth of x ∈ Rd;

Remark: Operator L is symmetric in L2(Γ, µ).Corresponding Dirichlet form is E−.

Questions

1 Existence of non-equilibrium dynamics (via general analytic approach)

2 Spectral properties (symbol)

3 Asymptotic behavior of dynamics

4 Speed of convergence to equilibrium

Harmonic analysis on configuration space

L : dFt

dt = LFt Γ, F oo〈F,µ〉=

∫Γ F dµ //M1

fm(Γ)

��

Lenard

L∗ : dµt

dt = L∗µt

L := K−1LK Γ0, G oo〈G, k〉=

∫Γ0Gk dλ

��

K(Γ0) L∗ : dkt

dt = L∗kt

K-transform:KG(γ) :=

∑ξbγ

G(ξ), γ ∈ Γ.

Construction of the process: main steps

The key ”function”: R(η) := C |η|, C > 0

Set of initial states

M1R(Γ) =M1

C(Γ) :={µ ∈M1(Γ) | kµ ≤ const · C|·|

Banach space (”densities” for locally finite measures)

KC := {k : Γ0 → R | k · C−|·| ∈ L∞(Γ0, λ)},

where λ := λ1 is the Lebesgue-Poisson measure with intensity 1.

Banach space (quasi-observables)

LC := L1(

Γ0, C|η|λ(dη)

Construction of the process: main steps

To show

1 Symbol generates C0-semigroup in LC ;

2 Growth condition: kt(η) ≤ const · C|η| (satisfied !!!);

3 Positive definiteness: preservation of the correlation function property.

The symbol of the operator (K- image) :

The formal K- image or symbol of the operator L:

L := K−1LK

Symbol of the Glauber dynamics with competition

L := L0 + L1 + L2,

whereL0G(η) := −A(η)G(η), A(η) =

∑x∈η

∏y∈η\x

eβφ(x−y);

L1G(η) := −∑

ξ⊂η, ξ 6=η

G(ξ)∑x∈ξ

∏y∈ξ\x

eβφ(x−y)∏y∈η\ξ

(eβφ(x−y) − 1);

L2G(η) := κ∫

G(η ∪ x)dx, κ > 0.

Banach space and domain of the symbol

We considerL : D(L) ⊂ LC → LC

in the Banach space

LC := L1(

Γ0, C|η|λ(dη)

The domain of L

D(L) := {G ∈ LC : A ·G ∈ LC} .

Existence of a semigroup

Proposition

For any triple of positive constants C, κ and β which satisfies

2eCst(β)C + 2κe2BβC−1 < 3

the symbol L is a generator of a holomorphic semigroup Tt, t ≥ 0 in LC .

Idea of proof

1 The operator (L0G)(η) := −G(η)∑x∈η

∏y∈η\x e

βφ(x−y) with

D(L0) =

{G ∈ LC

∣∣∣∣∣∑x∈η

eβE(x, γ\x)G(η) ∈ LC

is a generator of holomorphic semigroup;

2 The operator(L− L0, D(L)

)is relatively bounded w.r.t. (L0, D(L0));

3 The operator(L,D(L)

)is a generator of a holomorphic semigroup in LC

(Kato perturbation theory).

Existence of the process

Positive definiteness: approximation arguments

Result of the general construction approach:

Theorem (Kondratiev/K/Minlos)

For any µ ∈M1C(Γ), there exists a non-equilibrim Markov process with generator

L and initial distribution µ provided

2eCst(β)C + 2κe2BβC−1 < 3

holds.

Symbol: spectral properties

1 The point z = 0 is an eigenvalue of the operator L with the eigenvector

ψ0(η) ={

1, η = ∅,0, η 6= ∅.

2 There exists z0 > 0 such that

I1 = {z ∈ C : Rez > −z0} \ {0}

I2 ={z ∈ C : |arg z| < 3π

}\ {0}

belong to the resolvent set of the operator L.

Dual space

The dual rescaled space to the Banach space LC :

KC := {k : Γ0 → R | k · C−|·| ∈ L∞(Γ0, λ)}.

We consider alsoK≥1 := {k ∈ KC : k(∅) = 0}.

L∗ - the corresponding adjoint operator to the operator L on KCDuality:

� G, k �:=∫

G · k dλ1 =∫

G · C |·| · k

C |·|dλ1, G ∈ LC

Evolution on KC :� TtG, k �=:� G, T ∗t k �

kt := T ∗t k, t ≥ 0.

Evolution of correlation functions

T ∗t - the semigroup which corresponds to L∗ (in the weak sense)

K≥1 - invariant subspace

T ∗t = T ∗t |K≥1 restriction to the invariant subspace K≥1.

The semigroup T ∗t is such that for t ≥ 1

||T ∗t || ≤ const e−z02 t.

Idea of Proof

Space decomposition

L10 := {G ∈ LC |G = const ψ0}, ψ0(η) =

{1, η = ∅,0, η 6= ∅

and L≥1 := {G ∈ LC : G(∅) = 0}

LC = L10 + L≥1.

L10 is invariant with respect to the semigroup Tt;

Tt/L10

- factor semigroup on the factor space LC/L10

which can be identifiedwith the space L≥1

factor generator (L/L1

)[G] = L[G] := [LG],

where [G] ∈ LC/L10

is an equivalent class of the element G ∈ LC .

Idea of Proof

Similarity: L corresponds to the operator L11 := L|L≥1 in the sense ofsimilarity, i.e. there exists isomorphism J

J−1L11J = L/L10.

DefineTt := J Tt/L1

0J−1,

which generator coincides with L11.

T ∗t = T ∗t |K≥1 - adjoint semigroup to the semigroup Tt.

Idea of Proof

The inverse formula for a semigroup via resolvent

− 12πi

(L|L≥1 − z11)−2eztdz = tTt,

where integral is taken over the contour Υ of the form

Υ = {z = −z0/2 + iw : w ∈ R}

The resolvent bound∣∣∣∣∣∣(L|L≥1 − (−z0

2+ iw)11)−1

∣∣∣∣∣∣ ≤ 3

(e−4βB + w2)12

Ergodicity

Let initial measure µ0 ∈M1(Γ) has the correlation function k0 ∈ KC for someC > 0. Suppose that the parameters of our system satisfy the following conditions

exp {e2βBCst(β)C}+ 2κe2βBC−1 <32

κe2βBC−1 exp {e2βBCCst(β)} < 1

Denote kµ the correlation function corresponding to µ ∈ G(κ, β). Then,

||kt − kµ||KC≤ const e−

z02 t||k0 − kµ||KC

Condition 1: exp {e2βBCst(β)C}+ 2κe2βBC−1 < 32

1 Existence of the semigroup T ∗t in the Banach space KC2 Bound for the semigroup T ∗t

||T ∗t || ≤ const e−z02 t.

Condition 2: κe2βBC−1 exp {e2βBCCst(β)} < 1

1 Existence of µ ∈ G(κ, β)2 Bound for the corresponding correlation function

kµ(η) ≤ C|η|, η ∈ Γ0

rt := kt − kµ = T ∗t (k0 − kµ) ∈ K≥1 and as result

||kt − kµ||KC= ||T ∗t r0||KC

≤ const e−z02 t||k0 − kµ||KC

Thank you for attention! :)

glauber dynamics in continuum - uni-bielefeld.de

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