Grupo de Fisica Matematica Universidade de Lisboa

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The Renormalization of Self Intersection Local Times

I: The Chaos Expansion

~- Margarida de Faria, Custodia Drumond and Ludwig Streit

IFM 1/98

. -'. !"\ - ~1:lS- 'J 07 90 • FAX 35LLJ-'15..A288n~28~=.J....J.S.1

The Renormalization of Self Intersection Local Times

I: The Chaos Expansion

Margarida de Faria* Custodia Drumond*� Ludwig Streit*+�

*CCM, Universidade da Madeira, P 9000 Funchal� http://www.uma.pt/�

+BiBoS, Univ. Bielefeld, D 33615 Bielefeld�

June 17, 1998

1. Introduction

The intersections of Brownian motion paths have been investigated since the For­ties [20]. One can consider intersections of sample paths with themselves or e.g. with other, independent Brownian motions [36], one can study simple [5] or n-fold intersections [6] and one can ask all of these questions for linear, planar, spatial or - in general- d-dimensional Brownian motion: evidently self-intersections become increasingly scarce as the dimension d increases.

Intersection local times of Brownian motion were studied by many authors, see e.g. [1], [3]-[11]' [15], [18]-[38]. A more systematic review than we can give here of the subject will be found e.g. in the recent reference [15].

An informal but rather suggestive definition of self intersection local time of Brownian motion B is in terms of an integral over Dirac's - or Donsker's - 8­function

; L = Jd2t 8(B(t2) - B(t l )),

intended to sum up the contributions from each pair of "times" t l , t2 for which the Brownian motion B is at the same point.

- -

In Edwards' modeling of polymer molecules by Brownian motion paths, L is used to model the" excluded volume" effect: different parts of the molecule should not be located at the same point in space. As another application, Symanzik introduced L as a tool for constructive quantum field theory in [31].

A rigorous definition, such as e.g. through a sequence of Gaussians approx­imating the 8-function or in terms of generalized Brownian functionals [10] [30] [33], will lead to increasingly singular objects and will necessitate various "renor­malizations" as the dimension d increases. For d > 1 the expectation will diverge in the limit and must be subtracted [18] [32], clearly L will then no more be pos­itive. For d > 3,5,7, ... further subtractions have been proposed [33] that will make L into a well defined generalized function of Brownian motion.

For d = 3 another renormalization has been constructed by Westwater to make the Gibbs factor e-g

•L of the polymer model well-defined [35], for yet another,

recent approach see [1]. Yor, in [38], first suppresses the short time accumulation of self-intersections

by the regularization

8(B(t- 2) - -B(t1 )) ~.8(B(t2) - -B(t1) + c)

and shows, again for d = 3, that a multiplicative renormalization

gives rise to another, independent Brownian motion as the weak limit of regular­ized and subtracted approximations to L.

In the present note we study similar limits, for arbitrary d ~ 3, using a Gaussian regularization of the 8-function for which the chaos expansion of the corresponding regularized L e is available [10]. For a suitably subtracted and renor­malized local time, each term in this expansion converges in law to a Brownian motion.

We prepare and state these results in the following section 2, in section 3 we give their proofs. In a companion paper [3] we extend these results to the corresponding series.

2. Definitions and Main Results

2.1. White Noise Analysis and Local Times

We reproduce here some White Noise Analysis concepts as introduced in [10], referring to [14] for a systematic presentation.

2

Brownian motions Bi , i = 1, ... , d, have version in terms of white noise Wi via

Hence we consider independent d-tuples of Gaussian white noise w = (WI, ... ,Wd) and correspondingly, d-tuples of test functions f = (il, ... , Id) E S(R, Rd), and introduce the following notation:

d d

r;,= (nI, ... ,nd), n = L ni, r;" = IT ni! I I

d

< f, f > = L Jdt H(t) i=1

and similarly for <: w®n' :, Fn' > where for d-tuples of white noise the Wick product: ... : [14] generalizes to

- d w0ni. w0n '= JO,. .• . \CI. t ..

i=1

The vector valued white noise w has the characteristic function

C(f) = E(ei<w,f» = r dJ.L [w] ei<w,f> = e-~<f,f>, (2.1)j S·(R,Rd)

d

where < w, f > = I: < Wi, Ii> and Ii E S(R, R). i=1

The Hilbert space (L2

) = L 2 (dJ.L)

is canonically isomorphic to the d-fold tensor product of Fock spaces of symmetric square integrable functions:

(£2) ~(; SymL2(Rk, k!dkt))0d = J, (2.2)k=O

3

for the general element of (L2 ) this implies the chaos expansion

cp(w) = I:00

<: w®r; :, Fr; > (2.3)

with kernel functions F in J. It is desirable to introduce regularizations for the intersection local time, with

a view towards the construction of well-defined, "renormalized" intersection local times in higher dimensions where the latter do not exist without subtractions. A computationally simple regularization is, for c > 0,

with 1B 2be (B (t2) - B (t1)) =(21rc)-d/2 e_ (t );:(t,)I2 (2.4)

It has the following chaos expansion, which we quote here only for d ~ 3 :

Theorem 2.1. [10J For any c > 0, Le - E (Le) has kernel functions FE J given by

(2.5)

8(u)8(t - v) . ((v - u + c)-X: + (t + c)-X: - (v + c)-X: - (t - u + c)-X:)

ifall Ili are even, and zero otherwise, with V(Sl, ... , sn) =max(sl, ··'l sn), U(Sl, ... , sn) = min(sll '.'l sn), and x = (n + d) /2 - 2. 8 is the Heaviside function.

Each kernel function is thus the sum of four terms. The first one gives rise to

Definition 2.2.

Mt(d, rt, c) = r crs(v - U + c)-X: : w®n' (s) : (2.6)� J[O,tl n�

d� _ r crs(v - U+ c)-X: Q9 :wi(si) ...Wi(S~) :� J~~n i=l�

4�

1

d ni t Si d

- L L 1ds~ 1'" ~-lS(V - U + c)-X 0 :Wi(S;) ...Wi(S~J : i=l m=l 0 0 t=l

- t ni it dBi(T) iT ~-lS(T - U + c)-X : w®~(s) : i=l 0 0

d t

- L 1dBk(v)mk(V) (2.7) k=l 0

- Ld

Mk,t (2.8) k=l

The others give

Nt(d,n:,c) _ r dns((t+c)-X_(v+c)-X_(t-u+c)-X) :w®n(s): Jro,tjn

All of the above processes are continuous.

Definition 2.3. We denote the nth order contribution to the regularized local time Lf; by

Remark 1. Our key observation is that, as c goes to zero, 1\11 is more singular than N, and that it is a Brownian martingale.

2.2. The main theorems

Theorem 2.4. For d 2: 3, the renormalized M converge in distribution to inde­pendent Brownian motions {3i:

with 2 ---. { n(n - 1) if d = 3

kn = n! n!(d-4)! '[ d 3 (2.9) (n+d-5)! 1 >

5

while for d = 3

II(v - U + E:)-XII~2([o,tln) = n(n - l)t IInE:1 (1 + 0(1))

•To show convergence of these martingales by Theorem VIII.3.11 of [16] we

must show convergence of characteristics. Since the processes M are continuous this reduces to showing convergence in probability of (rI\lfi , rl'Vfk)t as E: -'> +0. This will be taken care of by proposition 3.6 which we prepare now:

(rl'Vfi,rlvh)t = r20ik it dv(mi(v))2

and we need to estimate

Let us note that

2E ((rMi, rMk)t) = E (r MiMk) = Oikr2 : k n! II(v - U + E:)-XII~2([o,t)n)

E ((rMi, rMk)t) = Oik nk k;' (t + 0(1)) n

Next we intend to show that the rhs gives indeed that the rest of (rMi, rl'Vfk)t goes to zero. The kernel of the highest order is

C(i) ( I. . ')( )-X( I )-X') -ltd 8(---+ ---+ Sl'Sl" ... ,Sn-l,Sn_l - T T-VVV T-U+E: T-U +E:2n -2 {j 0i

(3.1)

V(/) and U (I) are the largest and the smallest of the S~/), and (7!i) k == Oik.

(For n = 2: U(/) = v(1) = s(1)).

Remark 2. The integral (3.1) can be calculated in closed form (using e.g. {12} nos. 2.15 and 2.263.4), but one gets a useful approximation by introducing the following auxiliary functions:

H2n- 2 (Sl, ... , Sn-l; s;, ...,S~_l; E:; d) = (v V Vi - U V u' + E:)-(n+d)/2+3(v V Vi - U /\ u' + E:)-(n+d)/2+2

where v =max(si), U =min(si) and v' =max(s~), u' =min(sD. These functions majorize the kernel functions

7

Lemma 3.3. For n > 2, and x > 1

O G (i) (' I )< 2n-26i 81,81;, ; 8n -1, 8 n -1 (3.2)

< ~l H2n- 2(81 , , 8n-1; 8~, ... , 8~_1; e; d)x-

Proof:

o < ltdT8(T-vVVI)(T-u+e)-X(T-ul+e)-X

< it dT(T - U V u' + e)-X(v V v' - u A u' + e)-X vvv' 1 ­

< --(v V v' - u V u' + e)-x+l(v V v' - u A u' + e)-Xx-l

This estimate suffices to show

--+ --+Lemma 3.4. For 2 n - 28i =1= 0

r2G(~ __ -+ 0 in £2(R2n- 2)2 n -2 8 i

as e goes to +0.

Proof: Consider first n 2: 3. By the above estimate it is sufficient to show that

lim r4 II H2n- 211 ~2([0 t]2n-2) = o. e:->+O '

IIH2n-211~2(R2n-2) - J~-1 8 J~-1 8' (V V v' - u V u' + e)2-2x(v V V' - u A u' + e)-2x

t

du lvl

- Cn l dv l v

dv' l v

du'

(v _u)n-3(v'_u,)n-3

(V V v' - u V u' + e)n+d-6(v V v' - u A u' + e)n+d-4

< Cne8-2d I t

/e: dy lY

dy' lY

dx ly' dx'

1 (y - x V x' + 1)d-3(y - x A x' + 1)d-1

8

and then decompose the x-integration into the following two domains

X' < x and x < x'

In the first case

8 I e -

2d it/€: dy i Y

dy' i yl

dx' 1~ dx (y _ x + 1)d-3~y _ x' + l)d-l

8 /1- e -2d it/€: dy i

Y

dy' i yl

dx' (y _ X + l)d-l 1~ dx (y _ x ~ 1)d-3

We first estimate the last integral

y - x' if d = 3 In (y - x' + 1) if d = 41~ -(Y---X-~-l)-d--3 dx ~ { d~4 if d> 4

and one finds 0 (1) if d = 3

I = { 0(c7-2d) if d> 3

Hence, in both cases, r 4 I vanishes as e -- O. The second case is

t c l yl� 8 2d� / 1 1I = e - I dy lY dx ( )d-l l Y

dy' dx' ( I )d-3 (3.3) o 0 y-x+l x x y-x+1

Estimating the last integrand by 1 we find

Y y' ( )2

1d I 1 dx' 1 < y - x x y x (y - x' + 1)d-3 - 2

Substitution of these estimates into the integrals over x and y gives for d 2: 3

It/€: lY ( )2 if d ={ 0(1) 3

I :s const.e8-

2d dy dx ( y - X )d-l = O(c-lIne) if d = 4 o 0 Y - x + 1 O(c7 - 2d ) if d > 4

9�

so in that r 4 I vanishes as c -+ O. For n = 2 it is sufficient to use the estimate

o < G2 = it d78 (7 - U V u') (7 - u+ c)1-d/2(7 - U' + c)1-d/2

< it d7(7 - u V u' + c)1-d/2(u V U' - u /\ u' + c)1-d/2 = H 2(u, u') uVu'

and to verify that

•With this lemma we have established that the highest order term of the (renor­malized) quadratic variation goes to zero in quadratic mean for any t > O. The kernels Gk of the other terms are obtained by integrating over pairs of s, s' such as e.g. m

(i) .'• . ,it

Sym 0 dsG2(n_6.) (s, S, S2, S2""" Sn-l, sn_l)'

these new functions are in fact also bounded by an expression like (3.1), and hence by (3.2), so that for all c > 0,

With this goal we show

Lemma 3.5. Let n ;::: 2 and m > 1 , and let Fn,m be a function symmetric in the variables S and in the variables s', with

o < Fn,m(Sl, ... ,Sn-l,Sn,S~,.",S~_l'S~) (3.4)

< cit d78 (7- max (s, S')) (7- min (Si) + c)-m(7- min (s~) + c)-mo I~n I~n I~n

Then 3em < 00 such that

o < it dsFn,m(Sl,' .. ,Sn-l, S, s~, . .. ,S~_l' s) (3.5)

< em t d78 (7- n:tax (s, S')) (7- ~in (Si) + c)-m+l/2(7_ min (s~) + c)-m+l/2Jo I<n I<n I<n

10

l !

Proof: Under the assumption of the lemma

t

f dsFn,m (SI' ... , Sn-l, S, S~, ... , S~_I' S) :S o

t t

cfdsJ'dr8(r- max (Si'S~,S)) ( r- min (Si'S) + c) -m (r- min (S~, s) + c)-m O~t~n-l O~t~n-l O~t~n-l

o 0

Using u = min Si , u' = min S~

O~i~n-l O~i~n-l t

V = max Si Vi = max s' 09~n-l ' O~i~n-l t

(For n = 2: U(I) = V(I) = sf)). Assuming without loss of generality that u < u' we can decompose the s-integration of our estimate as follows

t t

C f ds f dr8 (r - max (v, Vi, s)) (r - min (u, s) + c) -m (r - min (u' , s) + c)-m

o 0

·8 (r - max (v, Vi, s)) (r - min (u, s) + c)-m (r - min (u' , s) + c)-m

- c j dr (J ds(r-s+€)-2m+ Jds(r-u+€)-m(r-s+€)-m�

vVv' 0 u�

+ "]' ds (r - u + €)-m (r - u' + €)-m + Jrds (r - u + €)-m (r - v! + e)-m)

~ vv~

We now show that each of the four terms obeys the postulated estimate (3.5)

t u t

Jdr Jds (r - S+ c) -2m :S 1 Jdr (r - u + c) -2m+l 2m-l

vvv' 0 vVv'

11

t

:::; 2m1_ 1 j d78 (7 - Vv V') (7 - u + C)-m+~ (7 - U' + C)-m+~

o

since u < u' and m > 1/2. The second term

t u'

j d7 j dS(7-U+c)-m(7-S+C)-m

vvv' u�

t�

< 1 jd7(7-U+C)-m(7-U'+C)-m+lm-1

vVv'� t�

< 1 j d78 (7 - VV V') (7 - u + c)-m+~ (7 - U' + c)-m+~ . m-1

o

using again u < u'. The third term

t vvv'

j d7 j ds (7 - U+ c) -m (7 - U' + c)-m

vvv' u'�

t�

- j d7 (7 - U + c) -m (7 - U' + c) -m (V V V' - u')

vvv'� t�

< j d7 (7 - U + c) -m (7 - U' + c) -m (7 - u')

vVv'� t�

< j d78 (7 - V V V') (7 - u + c)-m+~ (7 - U' + c)-m+~ .

o

Finally

t

j d7 j tds (7 - U + c)-m (7 - U' + c)-m

vvv' vvv'

12

tJdr (r - u + c;)-m (r - u' + c;)-m (r - v V v')

vVv'� t�

< Jdr (r - u + c;) -m (r - u' + c;) -m (r - u')

vVv'� t�

< Jdr8 (r - v V v') (r - u + c;)-m+~ (r - u' + c;)-m+~

• o

Combining this Lemma with the previous one we conclude that for all kernel functions Gk with k 2: 2 arguments

lim r411SymGkll2 ~ lim r411Gkll2 ~ lim r411Hkl12 = a e-++O e-++O e-++O

i.e. we have shown Proposition 3.6.

Proof of Theorem 2.4: The above limit is clearly (up to constants) the quadratic variation of a Brownian motion. Theorem 2.4 is then a consequence of e.g. Theo­rem VIII.3. 11 in [16], which in the present case of continuous martingales requires convergence in probability of quadratic variations for a dense set of t.

To control the remaining terms Nt(d, Tt, c;;) in the chaos expansion we observe that

IINt(d, Tt,c;)II~L2) - Tt! 11(t + c;)-X - (v + c;)-X - (t - u + c;)-XII~2([o,tJn)

< Tt! (II (t + c;)-XII~2 + \I (v + c;)-XII~2 + II (t - u + c;)-XII~2)

The first of these three norms is equal to tn(t + C;)-2x, i.e. 0(1). The second one IS

it / e 1t vn-l xn ­i4 dr ~s(v + C;)-2x = n dv = nc - dx-:---""'7""""'"~-:-

J[O,tl n o (v + c)n+d-4 0 (x + 1)n+d-4

0 (1) for d = 3 - O(lnc) for d = 4

{ 0(c4- d ) for d > 4

13

which are suppressed by the renormalization

1 r2(e) = {llne l - for d = 3.

ed- 3 for d> 3

A similar estimate holds for the third term of N, so that we have shown

Lemma 3.7.

ms - lim r(e)Nt(d, Tt,e) = O. e:--+O

In fact the convergence is uniform in any finite t-interval. Next we show

Lemma 3.8. The processes {r(e)N.(d, Tt,e;) : e > O}, {r(e)M.(d, Tt,e;) : e > O}� and their linear combinations are tight.�

Proof: A criterion for tightness of M (following [17], p.64) is�

sup E IrMt - rMs\O< S CT (t - s)1+13 , (3.6) e:>0

\IT> 0 and 0 ::; s < t ::; T and for some positive constants a, (3 and CT' As a first step we show

sup E IrMt - rMsl 2 S CT (t - s) (3.7)

e:>0

by direct calculation:

v2E IMt-M s l = Tt!n it dv l cr-1s (v - u + e)-2x

The second integral may be estimated as follows v� lv (v - u)n-2cr-Is (v - U + e)-2)( = (n - 1) du n+d-4

O o (v-u+e)

< (n - 1)c3-d lv/e: dx 1 d-2

lo (x + 1)

_� (n _ 1) { In (v + e) - In e for d = 3 O(e3-

d) for d> 3

14

Renormalization of this estimate by the factor r 2 makes it bOllllded on [0, T], and the integral over v gives the desired estimate (3.7), Le.

(3.8)

Note that the kernel functions of K are all dominated by those of M. Hence we have also, possibly with a larger constant CT, the estimate

(3.9)

By the hypercontractivity of the Ornstein-Uhlenbeck semigroup (see e.g. [14], p.235), one has for nth order white noise monomials r.p E (L 2 ), and any a > 2

For r.p = rK t - rK s ,and using the above estimate for the 2-norm, we get

E IrKt - rKsl Q

~ CT (t - st/2

as required to ensure tightness. The estimates for rN etc. are of the same kind.• Proof of Theorem 2.5: We need to consider

rK=rM+rN

knowing that, as c -+ +0, the rK are tight by Lemma 3.8, the rM converge in law, and the rNt go to zero in mean square. The latter two facts are sufficient, via the Cramer-Wold device (see e.g. [17] p.61), for finite dimensional convergence of rK; tightness then implies convergence in law.

Acknowledgements: This work has had partial support from PRAXIS XXI and FEDER. L. S. would like to express his appreciation for an inspiring discussion with L. Chen, for a very helpful remark of J. Potthoff, and for the splendid hospitality of the Grupo de Ffsica Matematica da Universidade de Lisboa, llllder the auspices of a "Marie Curie" fellowship (ERBFMBICT 971949).

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