A White Noise Approach to the Self Intersection Local Times of a Gaussian Process.

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J o u r n a l

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Vol. 20 No. 2

October 2014

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P u b l i s h e d b y t h e I n d o n e s i a n M a t h e m a t i c a l S o c i e t y

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A W h i t e Noise A p p r o a c h to the S e l f - i n t e r s e c t i o n L o c a l T i m e s o f a G a u s s i a n

Process

Herry Pribawanto Suryawan 111 -124

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V. V i l f r c d , A . Siiryakala, and K. Rubin M a r y 149-159

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V o l . 20, N o . 2 (2014), p p . 111-124.

A W H I T E N O I S E A P P R O A C H T O T H E

S E L F - I N T E R S E C T I O N L O C A L T I M E S O F A G A U S S I A N

P R O C E S S

HE R R Y PR I B A W A N T O SU R Y A W A N

Department of Mathematics, Sanata D h a r m a University,

Yogyakarta, Indonesia, herrypribs@usd.ac.id

Abstract. I n t h i s paper w c s h o w t h a t for a n y s p a t i a l d i m e n s i o n , t h e r e n o r m a l -ized self-intersection local times of a c e r t a i n G a u s s i a n process defined by indefinite W i e n e r integral exist a s H i d a d i s t r i b u t i o n s . A n explicit expression for the chaos de-c o m p o s i t i o n in t e r m s of W i de-c k tensor powers of w h i t e noise is also o b t a i n e d . W de-c also s t u d y a r c g u l a r i z a t i o n of the self-intersection local times a n d prove a convergence result i n the space of H i d a d i s t r i b u t i o n s .

Key words: b - G a u s s i a n process, w h i t e noise a n a l y s i s , s c l f - i n t c r s c c t i o n local t i m e .

Abstrak. D i d a l a m m a k a l a h ini d i b u k t i k a n b a h w a u n t u k s c b a r a n g d i m c n s i s p a s i a l r c n o r m a l i s a s i d a r i w a k t u lokal p c r p o t o n g a n - d i r i d a r i s c b u a h proses G a u s s i a n y a n g didefinisikan melalni integral W i e n e r tak t e n t u m e n i p a k a n d i s t r i b u s i H i d a . D e k o m -posisi chaos d a r i d i s t r i b u s i H i d a t c r s c b u t j u g a d i b c r i k a n s c c a r a c k s p l i s i t . S t u d i t e r h a d a p s c b u a h r c g u l a r i s a s i d a r i w a k t u lokal p c r p o t o n g a n - d i r i j u g a d i l a k u k a n d a n d i b u k t i k a n s c b u a h basil terkait k c k o n v e r g c n a n d a r i r c g u l a r i s a s i t c r s c b u t d i r u a n g d i s t r i b u s i H i d a .

Kata kunci: proses G a u s s i a n - b , a n a l i s i s w h i t e noise, w a k t u lokal p c r p o t o n g a n - d i r i .

1 . I N T R O D U C T I O N

As an infinite-dimensional stochastic distribution theory, white noise analy-sis provides a natural framework for the study of local times and self-intersection local times of Gaussian processes, see e.g. [4]. The concept of self-intersection local times itself plays important roles in several branches of science. For example, it is

2000 Mathematics Subject Classification: 60H40, 60G15, 28C20, 46F25. Received: 11-08-2013, revised: 05-04-2014, accepted: 19-05-2014.

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112 H.P. SURYAWAN

used i n the construction of certain Euclidean q u a n t u m field [ 1 0 ] . I n the Edwards polymer theory self-intersection local times appeared i n the p a t h integral to model the excluded volume effects of tiie polymer formations [3]. The first idea of analyz-ing local times and self-intersection local times usanalyz-ing a w h i t e noise approach goes back at least to the w o r k of Watanabe [ 1 1 ] . He showed t h a t as the dimension of the B r o w n i a n m o t i o n increases, successive omissions of lowest order chaos i n the W i e n e r - I t o decomposition are sufficient to ensure t h a t the t r u n c a t e d local t i m e is a w h i t e noise d i s t r i b u t i o n . A further investigation was given by D a Faria et al i n [1[. T h e y gave the chaos decomposition i n terms of W i c k tensor powers of w h i t e noise. T h e i r results were later generalized to fractional B r o w n i a n m o t i o n for any

H u r s t parameter

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

H 6 ( 0 , 1 ) by D r u m o n d et al [ 2 [ . I n the present paper we p r o -vide another direction of generalization of some results i n [ I ] to a certain class of

Gaussian process defined by indefinite Wiener integrals (in the sense of l t d ) .

First of all, lot us fix 0 < T < oo. T h e space of real-valued square-integrahle function w i t h respect to the Lebesgue measure on [0, T] w i l l he denoted by L^\0, T].

Let / e L ^ [ 0 , T [ and B = iBt)telo,T] be a standard onedimensional B r o w n i a n m o -tion defined on some complete p r o b a b i l i t y space ( n , F , P). I t is a fundamental fact

from I t o ' s stochastic integration theory t h a t the stochastic process

yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA

X = {Xt)te[o,T]

defined by the indefinite Wiener integral Xt := f f{u) clB^ is an L ^ ( P ) - continuous martingale w i t h respect to the n a t u r a l filtration of B. I t is also a centered Gaussian process w i t h covariance function E{XsXt) = /g*^* \ f{u)\^ du, s,t > 0, see e.g. [ 7 [ . Here E denotes the expectation w i t h respect to the p r o b a b i l i t y measure P. I n this work we further assume t h a t / is bounded and never takes value zero on [ 0 , T [ . We call the corresponding stochastic process as b-Gaussian process. B y choosing / to be the constant f u n c t i o n 1 , we see t h a t our new class of Gaussian processes contains B r o w n i a n m o t i o n as an example. Moreover, by d-dimensional b-Gaussian process we mean the r a n d o m vector {X^,...,Xf where X^,...,X'^ are d independent copies of a one-dimensional b-Gaussian process. M o t i v a t e d by similar works on self-intersection local times of B r o w n i a n m o t i o n and fractional B r o w n i a n m o t i o n , see e.g. da Faria et al [ 1 ] , D r u m o n d et al [ 2 ] and Watanabe [ 1 1 ] , we consider the

self-intersection local time of b-Gaussian process X, which is informally defined as

where 6 denotes the Dirac delta d i s t r i b u t i o n at 0 . The (generalized) r a n d o m v a r i -able ( 1 ) is intended to measure the amount of t i m e i n w h i c h the sample p a t h of a b-Gaussian process X spends intersecting itself w i t h i n the t i m e interval [ 0 , T [ . A p r i o r i the expression ( 1 ) has no inatiiematicai meaning since Lebesgue i n t e g r a t i o n of Dirac delta d i s t r i b u t i o n is not defined. One common way to give a m a t i i e m a t i -cally rigorous meaning to such an expression, as i n [I] and [ 2 ] , is by a p p r o x i m a t i o n using a Dirac sequence. More precisely, we interpret ( 1 ) as the l i m i t i n g object of the approximated self-intersection local t i m e Lx,e ( T ) of b-Gaussian process X defined as

(1)

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A W hite Noise Approach 113

as £ —> 0, where Pe is the heat kernel given by

1

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

( x^S

Pe{x) •- —f= exp -

— , X 6 K.

V 27r£ V 2£ /

This a p p r o x i m a t i o n procedure w i l l make the l i m i t i n g object, which we denote by L x ( T ' ) , more and more singular as the dimension of the process X increases. Hence, we need to do a renormalization, i.e. cancelation of the divergent terms, to o b t a i n a well-defined and sufficiently regular object.

Now we describe briefly our m a i n results. Under some conditions on the spa-t i a l dimension of spa-the b-Gaussian process X and the number of subtracted terms in the truncated Donsker's delta function, we are able to show the existence of the renormalized (or truncated) self-intersection local t i m e Lx (T) as a well-defined ob-ject i n some w h i t e noise d i s t r i b u t i o n space. Moreover, we derive the chaos decompo-sition of Lx{T) i n terms of W i c k tensor powers of w h i t e noise. T h i s decomposition corresponds to t h a t i n terms of multiple W i e n e r - I t o integrals when one works i n the classical stochastic analysis using Wiener space as the u n d e r l y i n g p r o b a b i l i t y space and B r o w n i a n m o t i o n as the basic random variable. Finally, we also analyze a regularization corresponding to the Gaussian a p p r o x i m a t i o n described above and prove a convergence result. The organization of the paper is as follows. I n section 2 we summarize some of the standard facts from the theory of w h i t e noise analysis. Section 3 contains a detailed exposition of the m a i n results and their proofs.

2. B A S I C S O F W H I T E N O I S E A N A L Y S I S

I n order to make the paper self-contained, we summarize some fundamental concepts of w i i i t e noise analysis used throughout this paper. For a more compre-hensive explanation including various applications of w h i t e noise theory, see for example, the books of H i d a et al [4], K u o [6] and Obata [9]. Let ( 5 ^ ( R ) , C , / i ) be the R'^-valued w h i t e noise space, i.e., 5^(]R) is tfie space of R'^-valued tempered distributions,yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA C is the Borel cr-algebra generated by weak topology on <S^(R), and the w h i t e noise p r o b a b i l i t y measure p is uniquely determined t h r o u g h the Bochner-Minios theorem (see e.g. [6]) by fixing the characteristic function

C { f ) : = / exp Ua, /)) dp{5j) = exp {-\\S\l

Js'gR) ^ ^ V 2

for a i l R'^-vaiued Schwartz test function / G iSd(R). Here |-|g denotes the usual norm i n the real H i l b e r t space L^(R) of all R"^-valued Lebesgue square-integrable functions, and (•, •) denotes the dual pairing between S'fR) and 5 d ( R ) . T h e d u a l pairing is considered as the bilinear extension of the inner product on L'^{R), i.e.

d

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114 H.P. SURYAWAN

for all .9 = ( f l i , . . . ,3d) 6

yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA

LUR) and / = ( / i , . . . , / d ) 6 Sd{R). We should remark t h a t we have the Gel'fand triple, i.e. the continuous and dense embeddings of spaces

SfR) ^ LliR) ^ S'fR).

Let / be a function i n the subset of L ^ [ 0 , T ] consisting ail real-valued bounded functions on [0, T ] w h i c h has no zeros. I n the w h i t e noise analysis setting a d-dimensioiial b-Gaussian process can be represented by a continuous version of the

stochastic process

X = {Xt)^^^Q J'^ w i t h

Xt:= ((•,

1( 0 , 1 1 / ) , . . . , ( - . l i o d i / ) ) , such t h a t for independent d-tuples of Gaussian w h i t e noise to = ( w i , . . . , W d ) G

s'dim

Xt{uj) =

( ( w i , l [ o , t ] / ) , . . . , (wd, l [ o , t ] / ) ) ,

where 1,4 denotes the indicator function of a set A C R .

Recall t h a t the complex H i l b e r t space L ^ ( / i ) : = L ^ ( 5 ^ ( K ) , C , / i ) is canonicaiiy u n i t a r y isomorphic to the d-foid tensor product of Fock space of symmetric square-integrable function, i.e.

L ^ ( M ) = 0 L ^ ( R ^ f c ! d % • ) , \fe=o /

via the so-called Wiener-Ito-Segai isomorphism. Thus, we have the unique chaos decomposition of an element F 6 L ^ ( M )

yvutsronmljihgfedcaVSQMJIE

i

Fiup,. . . , Old) = ^ f " ' ' ^ r ^ / ( m i , . . . , m . ) ) , (2) ( m i m d ) e N ; ;

w i t h kernel functions f(mi,...,ma) of the m - t h chaos are i n the Fock space. Here : w®'"^ : denotes the nij-th W i c k tensor power of ujj € 5](IR). We also introduce the following notations

d d

m = ( m i , . . . , m d ) G Ng, m-'Y'tnj, m ! = J) mf, 1 = 1

3 = 1

which simplify (2) to

Using, for example, the W i e n e r - I t o chaos decomposition theorem and the second quantization operator of the H a m i l t o n i a n of a harmonic oscillator we can construct the Gel'fand triple

( 5 ) ^ Lfp) ^

{sr

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of

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

[S) and ( 5 ) ' are also k n o w n as Hida test functions and Hida distributions, re-spectively.

The rest of this section is devoted to the characterization of a H i d a d i s t r i -b u t i o n via the so-called S-transforrn, wiiich can -be considered as an analogue of the Gauss-Laplace transform on infinite-dimensionai spaces. T h e S-transfonn of an element

4> G (S)' is defined as

where

( 5 $ ) ( / ) : = ( ( $ , : exp ( ( , / ) ) : ) ) , / G 5d(IR),

: exp ( ( , / ) ) : : = ^ ( : - ^ ^ :, Z ® ' " ) = G ( / ) exp ( ( , / ) ) ,

is the so-called W i c k exponential and ((•, )) denotes the dual pairing between (iS)* and {S). We define this dual pairing as the bilinear extension of the sesquilinear

inner product on L'^{p). The decomposition S^{f) = J^meN^ ^ E m , / ® ' " ^ extends the chaos decomposition to $ G (<S)* w i t h distribution-valued kernels such t h a t {{<!', (fi)) = Z^^ng^rf m ! ( F m . F m ), for every Hida test function ip G {S) w i t h kernel functions

yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA

ipm- T h e S-transform provides a quite useful way t o identify a Hida d i s t r i b u t i o n $ G {S)*, i n particular, when i t is very hard or impossible to find the explicit form for the W i e n e r - I t o chaos decomposition of $.

T h e o r e m 2.1. [5] A function F : SaiR) —> C is the S-transforrn of a unique Hida distribution in {S)' if and only if it satisfies the conditions:

( 1 ) F is ray analytic, i.e., for every f,g£ Sd{R) the mapping R 9 A i->

F ^ A / + 5 ^ has an entire extension to X £C, and

(2) F has growth of second order, i.e., there exist constants K\,K2 > 0 and a continuous seminorm ||-|| on SdfR) such that for all z eC, / G >Sd(R)

F{zf) < / G e x p f / G U P / "

There are two i m p o r t a n t consequences of the above characterization theorem. T h e first one deals w i t h the Bochner integration of a family of Hida distributions which depend on an additional parameter and the second one concerns tlie convergence of sequences of H i d a distributions. For details and proofs see [5].

C o r o l l a r y 2.2. [5] Let {U,A,ir) be a measure space and A $ A 6e a mapping from n to [Sy. If the S-transform O/4>A fulfds the folloviing two conditions:

(1) the mapping A i-> 5 ( $ A ) ( / ) is measurable for all f G «Sd(R), and

(2) there exist C i ( A ) G L^{n,A,ir), C2(A) G L°°{n,A,u) and a continuous seminorm \\-\\ on Sd{R) such that for all z £C, f£ Sd{R)

(10)

116 H . P . SURYAWAN

then <I>A is Bochner integrable with respect to some Hilbertian norm which topolo-gizing

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRPONMLKJIHGFEDCBA

( 5 ) * . Hence j^^\dv{\) G ( 5 ) * , and furthermore

s( f C F A I M A ) ) = / 5( $ A) r M A ) .

\Jn J Jn

C o r o l l a r y 2.3. [5] Let (<hji)„g[^ be a sequence in {S)' such that

( 1 ) for all f G 5 d( R ) , ( 5 ( $ „ ) ( / ) ) is a convergent sequence in C, and

( 2 ) there exist constants Ki, K2 > 0 and a continuous seminorm \\-\\ on SdfR) S{^„){zf) < Ki exp

(K

2\Z\^

yvutsronmljihgfedcaVSQMJIE

/ I

^], for all

2

G C, / G 5 d( R ) ,

such that

n G N .

Then {^n)neN converges strongly in {S)' to a unique Hida distribution <I> G ( 5 ) * .

3. M A I N R E S U L T S

I n several applications, we need t o " p i n " a b-Gaussian process at some point c G R*^. For this purpose, we consider the Donsker's delta function of b-Gaussian process which is defined as the informal composition of the Dirac delta d i s t r i b u t i o n

6d £ ^^R"^) w i t h a d-dimensionai b-Gaussian process (X()jg[o7-], i.e., Sd {Xt - c). We can give a precise meaning t o the Donskers's delta function as a H i d a d i s t r i b u -t i o n .

P r o p o s i t i o n 3.1. Let X = {Xt)t£io,T] be a d-dimensional b-Gaussian process and

c G R*^. The Bochner integral

( 1 \ V 2 7 r ;

7

6d [Xt -X,-c) :=

is a Hida distribution with S-transforrn given by

S{6d{Xi-Xs-c)){f)

I 1

e x p ( i A ( A t - X , - c ) ) dA, tfs

(3)

s A t /

exp Y[ fj(u)f{u)du-c,

\

for allf= ( / i , . . . , / d ) G 5 d ( R ) .

P R O O F . W i t h o u t loss of generality, we may assume t > s. Define a m a p p i n g F A :

5 d( R ) ^ C by F A : = S (exp {iX{Xt - X, - c ) ) ) . T h e n we have for any / G Sd{R)

(11)

= exp / 1

2 / ^ exp

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

i-iXc)

exp ( ( w , i A l[ s , t ] / +

fj^ dp{Q)

= exp [ - \ | A p l / M p d n ) exp (zA ( ( / , - c ) ) .

The mapping A i-4- % ( / ) is measurable for all / e 5 d ( R ) . Furthermore, let

yvutsronmljihgfedcaVSQMJIE

2 G C and / G

SfR), then

rt

Fxizf)

< exp ( - ^ | A p r\f{u)fdu\exp ([AjUl | ( / , l [ . , q / ) |) < exp ( - - I

< exp I - ^ 1

|/(u)p du ) exp | 2| / ? ( t- 6' ) ^ s u p | / , ( n ) | / ( n ) p d n exp

\rjf{uwdu

< e x p ( - l | A | V ( t- 5 ) ) exp for some positive constants a and f, and || • defined as

d

| | / | | o o : = $ 7 s u p | / , ( K ) | .

is a continuous seminorm on Sf

1 = 1

The first factor is an integrable function of A, and the second factor is constant with respect to A. Hence, according to the Corollary 2.2 dU {Xt - X^ - c) £ ( 5 ) * . Now, we integrate F A over Rf to obtain an explicit expression for tfie S-transforrn.

S{bd{Xt-X,-c)){f)

= ( y ^ )

S

(exp

{iX{Xt

- X , - c))) ( / ) riA

271

271 V^'AL (U{!lfj{u)J{u)du-c,))

I

exp

2 \

2 7 i/ ; | / ( u ) P d n

\ D2 /

exp

/

\ 2 / ; i / ( i i ) P r f i x^ t t

2 / ; | / ( i z ) | 2 d n

ffu)J{u)du-c,

2\

I

(12)

simplicity, we also take c = 0. Moreover, we define the truncated exponential series

e x p ( ^ ) ( x ) : =yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA Y

-and the truncated Donsker's delta function

AM

sT'i^t-xf

via

5 ( < 5 f ) ( X , - X , ) ) ( / )

1

2 ^ / ; i / ( n ) | 2 d u , exp

(N)

2 / .

T-^

Y

( t fjiu)fiu) du]

l\f{u)?dujr{\Js J

for every / G <Sd(R). Using Theorem 2 . 1 , one can verify easily t h a t d^^^ ( X t - X^,) is a well-defined element from

( 5 ) * .

T h e o r e m 3.2. Let X = (Xf)jgjQ j,j be a d-dimensional b-Gaussian process. For any pair of integers d> I and N >0 such that 2N > d - 2, the Bochner integral

L'PiT)

: =

jjT

{Xt-Xf

d\s,t)

is a Hida distribution.

P R O O F . From the definition of the truncated Donsker's delta function we see immediately t h a t S (Xt - Xs)j ( / ) is a measurable function for every / G

S d { R ) . Furthermore, for every

yvutsronmljihgfedcaVSQMJIE

2 G C and / G 5 d( R )

<

{Xt - Xf) {zf)

( 1

YV

Y'^

\2nfl\f{uWdu) exp

( 1

v

d/2 ₁

27ra2 ) j t -( 1 j-^i^

(xV"

exp

2 / ; | / ( n ) | 2 d n '

d^ n - I 2 \

2 \

f

O O

2 a 2

27ra2 ( t - s O ^ - ' ^ / ^ e x p f ^ l z P V 2Q2 o o /

2

(13)

A White Noise Approach 119

P r o p o s i t i o n 3.3. L e i X = (Xt

yvutsronmljihgfedcaVSQMJIE

)jgjQ

be a d-dimensional b-Gaussian process. For any pair of integers d>l and N > 0 such that 2N > d - 2, the kernel functions

F2m of ^Y {T) are given by

L2m(wi, • • • ,'"2m) = {-\r (1 Y "

yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA

f n ? r i

iVM

( i M/ ) ( ^ o m! V27r,

/' [[i=i\---is,t\j J y-ii ,2/ ( / : i / ( a ) P d « )

for each m £ NQ such that m> N. All other odd kernel functions Fm vanish.

P R O O F . Let / = ( / i , . . . , ff e SfR). The S-transform of L ) ^ ' ( T ) is obtained as follow:

1

S

YV

Y_{^x^'iT)) if) =}_1^

(2nJl\f{uydu)

d/2

I

X exp {N)

\

2 / ; i / (

YY

dfs,t)

[rjfiuWdu)

mj,...,md ^ j=l Ms J

mi+...+md=m

Remember the general form of the chaos decomposition

FTY)= Y (••^^'"••:Fm) and sYx^\T)){f)= Y {Fm, f^""

m+d/2

2m i

meN: '0

Hence, we can read ofi' the kernel functions F ^ for L ^ Y Y ) :

F , . = ^

m ! V27r/

IA f ft

®2m

Y

'lfiuWdu)

m+d/2

d^(s,t).

More precisely, for every m G Ng such that m> N and u i , . . . , U2m G R i t holds

F2m{u\,.

.

.,U2m) =

m ! V 2 W 4 ( j ; | / ( . ) | 2 , , )

while ail other odd kernels are identically equal to zero. •

m+d/2 dfs,t),

(14)

120 H . P . SURYAWAN

times L

yvutsronmljihgfedcaVSQMJIE

) 7'

{T) which is given by

E . ( L 4 m ) = Fo =

7)''7A

7 ^ ^ - 4 7 ^

It is immediate t h a t the generalized expectation is finite only in dimension one, and for higher dimension (d > 2) the expectation blows up. Now we extend the result i n Theorem 3.2 to local times of intersection of higher order m € N . The basic m o t i v a t i o n for this investigation comes from the situation when we want to count the amount of time i n which the sample p a t h of a b-Gaussian process spends intersect itself m-times w i t h i n the time interval [0, T ] . The following theorem gives a generalization to a result of Mendonca and Streit [8] on Brownian motion.

T h e o r e m 3.4.

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Let X = {Xt)t^[Qg\ be a b-Gaussian process. For any pair of integers d > 1 and N > 0 such that 2N > d —2 the (truncated) rn-tuple intersection local time of X

4 E m - AM ( % , - X , J. . . 5 4( X , , „ - X , , „ _ , ) d - t , (4)

where A , „ : = { ( t i , • • • , t,„) e R™ : 0 < G < • • • < t,„ < T } and d'^t denotes the Lebesgue measure on A „ , , is a Hida distribution.

P R O O F . Let

m - l

fc=l

Im:=l[^d''\Xt,,,-Xty

k--denote the integrand i n (4). Then

1

m - l / /

s

{im

)u)

=n

fc=l \ \

\ d/2

X exp

2 7 r / ; ; - i / ( u) | 2 d R y /

(M 1

v

f3{u)f{u)du

2\ \

which is a measurable function for every / G 5 d ( R ) . Now, we ciieck the boundedness condition. Let 2 G C and / G «Sd(R). Let us also define a continuous seminorm Hoo.fc o n 5 ( R ) by l / j U . f c : = sup„g(t^,t,,^,] |/j(u)| and a continuous seminorm |M|oo,fc on 5rf(R) by ||/||^_, : = E •=! UMo.k- Then, we have

\ d/2

m-\ I ( s{im){zf)\ = n

fc=i V

1

\2rrl\Y\f{uydxy

X exp kM

2 / ; - ' | / H | 2 d n \f{u)\du

(15)

A W hite Noise Approach

m - l

fc=i 27ra2(G.

X d / 2

exp {N) / m - l

121

\

\ f c = l

Finally, by defining another continuous seminorm || • ||» on <Sd(]R) by

4 = E

l c = l

114

we o b t a i n t h a t

S{Im){zf)

<

(1 (^-O

d / 2

[TJ

N

\

{tk+1

- t - tk) ,.y-D2 exp ( p k P l l / l P ) ,

for some p > 0. To conclude the proof, we notice t h a t for X — d/2 + 1 > 0 the coefficient i n the front of the exponential is integrable w i t h respect to d ' " t , t h a t is

/• Y\(f t xN-dn,m, ( r ( i + / v - r f / 2) r - '

7 l[ "''^ r ( m + l + ( m - l ) ( i V - d / 2 ) ) '

where r ( - ) denotes the usual G a m m a function. Therefore we may apply Gorollary 2.2 t o establish the existence of the Bochner integral asserted i n the theorem. •

To conclude the section, we present a regularization result corresponding to the renormalization procedure as described in Theorem 3.2. We define the regularized Donsker's delta function of b-Gaussian process as

<3d,e{Xt - Xs) ••= 1 \

27Te

d / 2

exp

V

\Xt - X , 2£

and the corresponding regularized self-intersection local t i m e of b-Gaussian process as

LxAT) := / 5dAXt-Xfdfs,t).

J A

T h e o r e m 3 . 5 . Let X = (Xt)tg[o7-] he a d-dimensional b-Gaussian process. For all e > 0 and d > 1 the regularized self-intersection local time Lx,e{T) is a Hida distribution with kernel functions in the chaos decomposition given by

F e, 2 m ( M l , • • • , M 2 m ) =

m ! m + d / 2

dfs,t)

for each m e Ng, and F j . m is identically to zero if m is an odd number. Moreover, the (truncated) regularized self-intersection local times

LYAT) : = JjZ^ {Xt-Xf d\s,t)

converges strongly as £ —> 0 in ( 5 ) * to the (truncated) local times L^YiY, provided

(16)

P R O O F . T h e first part of the proof again follows by an application of Corollary 2.2 w i t h respect to the Lebesgue measure on A . For all / € 5,i(R) we o b t a i n

s{Sd,AXt-xf){f)

/

1

\ d

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

/2

2^ ( e + / ; i / ( u ) P d i z )

X exp •Y( f Mu)f{u)du

i = l ^-^^

2\

2(f + Jl\f{u)\^du) U

which is evidently measurable. Hence for all

yvutsronmljihgfedcaVSQMJIE

2 6 C we have

S{6d,AXt-Xf){zf)

<

^d/2 I

exp

eY\l\!(u)\^du)} \ 2 ( e + / ; | / ( K ) P d n ) (t - sfA

7

\ d/ 2

IS i n -We observe that , A, j K i , j is bounded on A and ( — 7— - - , , 1.

| / ( " ) P d u \^27r(£ + F | / ( u ) P d u ) J

tegrabie on A . Hence, by Corollary 2.2, we have that Lx.fT) G ( 5 ) * . Moreover, for e v e r y / G 5 d ( R ) ,

d / 2

siwAmn-A) LY

_V

YV

V

1

[e + Sl\f{uWdu)

m+d/2

, d p pt .2mj

E

j A l i i

fAAAAdu] dfs,t).

m i , . . . , m d j - i V

mi + ...+md=m

I t follows from tlie last expression t i i a t the kernel functions F^ ^ m appearing i n the chaos decomposition

LxAT)= E {••^^^'"••'Ym)

are of the form

F e, 2 m( w i, - - - , R 2 m ) =

m! V 2 ^ y

[e + fAfiuWdu)

m+d/2

dfs,t),

and are identically equal to zero if m is an odd number. Finally we have to check the convergence of L^ l i T ) as e ^ 0. For all 2 G C and all / G Sd{R) we have

(17)

A White Noise Approach 123

<

d / 2 . ^ s

exp TA_

2

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

E

2 \

giving the boundedness condition. Furthermore, using similar calculations as i n the proof of Theorem 3.2 we obtain t h a t for ail (s, t) e A

s

{Xt - Xf)

if)

d/2

\2ir Z\f{u)\^du exp

(N)

<

f

1 \

27ra2

d / 2

, 2 / ; i/ ( w ) P d n

/

2 \

/

f]

_ « l Y - d / 2 exp

YT

2Q2

2 \

o o /

The last upper bound is an integrable function w i t h respect to d 2 ( s , t). Finally, we can apply Lebesgue's dominated convergence theorem to get the other condition needed for the application of Corollary 2.3. T i l l s finisiies tiie proof. •

4. C O N C L U D I N G R E M A R K S

We have proved under some conditions on tiie number of divergent terms must be subtracted and the spatial dimension, t h a t self-intersection local times of d-dimensionai b-Caussian process as well as their regularizations, after appropriatelly renormalized, are H i d a distributions. The power of the method of t r u n c a t i o n based on tiie fact t i i a t tiie kernel functions i n tlie chaos decomposition of decreasing order are more and more singular i n the sense of Lebesgue integrable function. F x p i i c i t expressions for the chaos decompositions of the self-intersection local times are also presented. We also remark t h a t white noise approach provides a general idea on renormalization procedures. T h i s idea can be further developed using another tools such as M a i i i a v i n calculus to obtain regularity results.

R E F E R E N C E S

[1] d a F a r i a , M . , H i d a , T . , S t r c i t , L . , a n d W a t a n a b e , H . , " I n t e r s e c t i o n local t i m e s as generalized w h i t e noise f u n c t i o n a l s ' " , Acta Appl. Math. 4 6 (1997), 351 - 362.

[2] D r u m o n d , C., O l i v c i r a , M . J . , a n d d a S i l v a , J . L . , " I n t e r s e c t i o n local t i m e s of fractional B r o w n i a n m o t i o n s w i t h H G ( 0 , 1 ) as generalized w h i t e noise f u n c t i o n a l s " . Stochastic and Quantum Dynamics of Biomolecular Systems. AIP Conf. Proc. (2008), 34 - 45.

[3] E d w a r d s , S. P . , " T h e s t a t i s t i c a l m e c h a n i c s of p o l y m e r s w i t h e x c l u d e d v o l u m e " , Proc. Phys. Soc. 8 5 (1965), 613 - 624.

[4] H i d a , T . . K u o , H - H . , Potthoff, J . , a n d S t r c i t , L . , White Noise. An Infinite Dimensional Calculus, K l u w c r A c a d e m i c P u b l i s h e r s , 1993.

[5] K o n d r a t i e v , Y . , L e u k e r t , P., Potthoff, J . , S t r e i t , L . , a n d W e s t e r k a m p , W . , " G e n e r a l i z e d functionals in G a u s s i a n s p a c e s : T h e c h a r a c t e r i z a t i o n t h e o r e m r e v i s i t e d " , J. Funct. Anal. 1 4 1 (1996), 301 - 318.

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124

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

H . P . SURYAWAN

[7] K u o , H - H . ,

Introduction to Stochastic Integration, S p r i n g er V c r l a g , H e i d e l b e r g , 2006. [8] M e n d o n c a , S. a n d S t r c i t , L . , " M u l t i p l e intersection local t i m e s in t e r m s of w h i t e n o i s e " .

Infinite Dimensional Analysis, Quantum Probability and Related Topics 4 ( 2 0 0 1 ) , 533 - 543. [9] O b a t a , N . , White Noise Calculus and Fock Space, S p r i n g e r V c r l a g , H e i d e l b e r g , 1994, [10] S y m a n z i k , K . , " E u c l i d e a n q u a n t u m field t h e o r y " , R . J o s t ( e d ) . Local Quantum Theory, A c a

-d e m i c P r e s s , N e w Y o r k , 1969.