AND WORDS

VOL. 21 NO. (1998) 73-78

A CHANGE OF SCALE FORMULA FOR WIENER INTEGRALS OF CYLINDER FUNCTIONS ON ABSTRACT WIENER SPACE

YOUNG SIK KIM

GLOBAL ANALYSIS RESEARCH CENTER SEOUL NATIONAL UNIVERSITY

SEOUL

151-742,

KOREA

(Received

^April

24,

1996 and in revisedform

August 12, 1996)

ABSTRACT

The purpose of this paper is to establish the existenceofanalytic Wiener and

Feynman

integrals foraclassofcertain cylinder functions whichisof the form

F(x) f((h,,x)’,... (h,,,x)’),

x

. ^B,

ontheabstractWiener space, and to establish therelationship betweenthe Wiener integral and the analytic

Feynman

integral forsuch cylinder functions on the abstract Wiener space.

We

then establisha changeof scale formula for Wiener integrals of such cylinder functionson the abstract Wiener space.

KEY WORDS AND PHRASES

Wienermeasure,analytic Wiener integral, analytic

Feyn-

manintegral,changeof scaleformula.

1991

AMS SUBJECT CLASSIFICATION CODES

Primary28C20.

1.

INTRODUCTION

In [4], ^Cameron

and Storvickexpressed the analytic Wiener and

Feynman

integralsas the limitsof Wiener integrals for certain Banach algebra

,S(L’[a,b])

of functionals. Using these results, they found a rather nice change of scale formula for Wiener integrals on a classical Wiener space

[5]. In [13;14], Yoo, Yoon

and Skoug extended these results to an Yeh-Wiener space and toanabstract Wiener space.

In [13], Skoug

and

Yoo

expressedtheanalyticWienerand

Feynman

integralsasthelimitsof Wiener integrals, and thentheyestablished achange of scaleformula forWienerintegralson theFresnel class of the abstract Wiener space.

In

this paper,we will show that the analytic Wiener and

Feynman

integrals ofcertaincylinder functionsontheabstract Wiener spaceexist,andwewill establishthe relationship between the Wienerintegraland the analytic

Feynman

integralfor suchcylinderfunctionsonthe abstract Wiener space. Usingthese

results,

wewill establishachange of scale formula forWienerintegrals of suchcylinderfunctionsonthe abstract Wienerspace.

Note

that the Fresnel classontheabstractWienerspaceconsistsof bounded functionsbut notallcylinderfunctionsareboundedingeneral.

2.

DEFINITIONS AND PHRASES

Let H

bearealseparableinfinite dimensional Hilbert space withinnerproduct

(., .>

^andnorm

I" ^(V/’, "> ^{Let I1" [10}

^{beameasurablenormon}

^H

^{withrespectto the}

^Gauss

^measure#.

^{Let B}

denote thecompletion of

H

withrespectto

I1" II0. ^Let

denote the naturalinjection from

H

into

Typeset

by

74 Y.S. KIM

B.

The adjoint operatori" of isone-to-one and maps

B"

continuously ontoadense subsetof

H’,

where

H"

and

B"

aretopologicaldualsof

H

and

B,

respectively.

By

identifying

H

with

H"

and

B"

withi’B’,^wehaveatriplet

(B’,H,B)

^{such that}

B"

C

H" H

C

B

and

(h,z) (h,x)

for all h in

B"

and zin

H,

where

(-,-)

denotesthe naturaldual pairing between

B"

and

B. By

awell known resultof

Gross [8;12],

#. ^-1 has auniquecountablyadditive extension v tothe Borel a-algebra

B(B)

on

B.

The triplet

(B,H,v)

is calledan abstract Wiener space and v is calleda Wiener measure.

We

denote the Wiener integralofa functional

F

by

fs ^F(z)v(dx).

For

moredetailssee

[8;12].

Let {e}=

^{denote a}complete orthonormal system in

H

such that

e’s

^{are in}

^{B’. For}

^each

h

H

and z

B,

wedefineastochasticinnerproduct

(-,-)~

between

H

and

B

asfollows:

(h,z) ,hrn,, _. (h,%)(e,z),

if the limit exists,

(2.1)

0, otherwise.

It

is well known

[11]

^{that for every h}

H, (h,x)

^{exists for v-a.e, x in}

B

and it hasa Gaussian distribution with meanzeroandvariance

Ihl -. Furthermore,

^{it is}easy to show that

(h,x) (h,x)

^{forv-a.e, x in}

B

ifh E

B, (h,x)

is essentiallyindependent of thecomplete orthonormal set used in itsdefinition,andfinallythat if

{h,... h}

^isanorthonormal setof elements in

H,

then

(h,x)~, (h,x)

areindependentGaussian functionals withmean zero andvariance one.

Note

that if bothhand xarein

H,

then

(h,x) (h,x).

Throughoutthis paper, let

R

denote the n-dimensional Euclidean space and let

C, C+,

and

C

^{denote the}complex numbers,thecomplex numberswith positive realpart,and thenon-zero complexnumbers with nonnegative realpart, respectively.

DEFINITION

2.1

Let (B, H,v)

be an abstract Wiener space.

A

function

F

is calleda cylinder

function

^on

B

if thereexists afinitesubset

{g,-..

g

}

^of

^H

^{such that}

F(x) ((g,x)~, (g,x)~),

x

B, (2.2)

where is a complex-valued Borel measurable functionon

R

k.

It

is easy to show that there exists alinearlyindependent set

{hi,-.. h}

^of

^H

^suchthat every cylinder function

F

of the form

(2.2)

^isexpressed^as

F(x) f((h,x)~, (h,,x)~),

x

B, (2.3)

where

f

^isa complex-valued Borel measurable functionon

R

". Thus we loseno generality in assuming that everycylinderfunctionon

B

isof the form

(2.3).

DEFINITION

2.2

Let F

be a complex-valued measurable function on

B

such that the integral

J(F; A) ./ F(A-1/2x)v(dx) (2.4)

exists forall real

A >

^0. Ifthere existsafunction

J*(F; z)

analyticon

C+

^{such that}

^{J(F; A)} J(F; A)

^{forall real}

^A > 0,

thenwe define

J*(F; z)

^{to be theanalytic} Wiener integralof

F

over

B

withparameter z, andfor each z

C+,

^wewrite

I"(F; z) J(F; z). (2.5)

Let

qbe a non-zeroreal number and let

F

beafunctionon

B

whose analytic Wienerintegral existsfor eachz in

C+. ^If

the followinglimitexists, then^wecallitthe analytic

Feynman

integral of

F

over

B

withparameterq, andwewrite

I1(F;q)=

lim

I(F;z), (2.6)

z---sq

where zapproaches-iqthrough

C+

^{and -1.}

DEFINITION

2.3

Let (B, H, v)

beanabstract Wiener space.

Let

nbeapositive integer, and let

{h,-.. h}

^beanorthonormal set of elements in

H. For

1

<

^p

^{< x,}

^let

^{’(n; p)}

^denote

theclassof cylinder functions

F

withthe formasfollows:

F() f((h,)~,..., (h,)~), e ,

where

f

^]R

^C

^isin

Lp(]R),

^{the spaceoffunctions}whose p-thpowersareLebesgue integrable onllano

Let ’(n; o)

^{denote theclass of}cylinder functions

F

with the formasfollows:

F(x) f((hl,X)~, (h,.,,x)"),

^x

B, (2.8)

where jr

R

Cis in

C0(R"),

the spaceofcontinuousfunctionson

R"

thatvanishatinfinity.

We

will close this section by mentioning the following useful theorem which is called the WienerIntegrationFormula.

THEOREM

2.4

Let (B,H,v)

be an abstract Wiener space and let

{h,---,h}

be an orthonormal setof elementsin

H. Let F B

Cbeafunction definedbytheformula

F() ((h,,)~, (h,)~), ,

where

f ^:JR

^{C isaLebesgue} measurable function. Then

/zF(x)’(dx)=/Bf((h,x)","", ^{(h, x)~) v(dx)}

(-) f()-exp{-[l ^{}d, (2.9)}

where

(u,,.--, u) R", [[2 E3=I ’/"t32,

^and

^d

^du..,du,.,.

In

the next section,wewilluseseveraltimesthefollowingwell-known integration formula:

exp{-u + i}

^du

exp{- yg ^{}, (2.10)}

where is acomplex numberwith

Re > O,

^{b isareal}

number,

and -1.

3.

THE MAIN RESULTS

We

willbeginthissectionby showingthatthe analytic Wiener integral of

F

exists for every

F _<p<_’(n; p)

andthat theanalytic

Feynman

integralof

F

existsforevery

F Y(n: 1).

THEOREM

3.1

Let (B,H,v)

^{be anabstract Wiener}space and let

F ’(n;p)

be given

y (.)

^o:

(.s),

wh

< < .

^The.:

(i)

the analytic Wiener integrals of

F

exist, andfor every z

C+,

z

.) ^/()exp{_Z

I""(F; z) (-) 5[[2} ^{d, (3.1)}

(ii)

for everynon-zeroreal numberq, and for

F ’(n; 1),the

^analytic

^Feynman

integral of

F

existsandisgiven

by

iq iq

i-l: ^{} d- (3.2)}

I’I(F; q) (-r) /() ^exp{-

and

d- du

^du,.,.

h

(,,..., ) e n , I1 := ,,

76 Y.S. KIM

PROOF. By

Theorem 2.4,we havethat for all real

A >

0

J(F; A) =/, F(A-1/2x)v(dx)

(-)- J’()exp{-l’l ^2}

Let ’(F;) () f l()exp{-ll}d,

^z

e C+.

^Then

^{J’(F;A) J(F;A)}

^{for all reel}

A>O.

e

ilu

orer’s

Theoremtosbo

tt J’(F; z)

ⁿ

anc

^{[unctono[ z}

ⁿ +.

^Fkst

o[

I,

bythe Dominated

Convergence Theorem,

ecn

so

tbt

J’(; z) s

continuouson

C+;

an

pproprte

dominating function

s

obtainedalmostexactly

n

thefo1og gument.

No et

be anyrectabe

smpe

codcurve

yng n +. ^e

nd only sbo that

J’(F; z)

^{d 0.}

But

thiswill clearly follow from the Cauchy

Iteal

^{Theorem if we canjustify}

movg

the line

teal

^along

r

inside the other

inteals

^defining

J’(F;). Let sup{li

and

inf{Rez e ft.

^If

^F

^{belongs to}

^{(n; 1),}

then the funetio

()lI()l

^dominates

() II()lexp{-ll }

^ad

iteable

^on

^N .

^If

^F

^belongsto

^{(;p) (1 <}

^p

^{< ),}

^the

the function

()lI()lexp{-l }

dominates

()lI()lexp{ -11 ^}

^dis

^iteeble

on

N

by H61der’s Inequality. If

F

belongsto

(n; ),

then the fuctio

() ^M p{-}

aomiates

() II()l exp{- }

^adis

inferable

^on

^N,

^where

^M

^aboundwith

M

forall

N .

Hence

we canapplyubini’sTheoremto the

teel fr ^{J’(F; )}

^{d and thenwehave}

fr ^J’(F;

d 0, becau the functio

() exp{- }

^analyticon

C+.

^Thenwehave established

(a.1).

Finallytheproofof

(a.2)

immediate.

In

order toobtain our mainresults, wend thefollowinglemma:

LNMMMN

^g.

t (B, H,u)

be abstract Wiener space and let

{h,,..., h}

be Definition 2.a.

Let F e (n;p)

be given by

(2.7)

^or

(2.8),

where 1 p

.

^{Then forevery}

+,

the functional

2 isWiener

inteable

^on

^B.

PNOO. By

Theorem

2.4,

wehave thatfor every

C+,

( )

[(h,, ^{)1,}. F()()}

"

1

zl ^{} d}

=()= .Y()"e{-

d

d _du...du.

We

canshowthat the lt

teal

^h:afinite value byusing thesameargument inthe

proof

of Theorem 3.1. Thusthe

proof

ofthislemma complete.

THEOREM

3.8

Let (B,H,)

^{be anabract Wiener}space and let

{h,,..-,h}

^be

Definition2.3.

t F e Y(n;p)

begivenby

(2.7)

^or

(2.8),

^{where 1}

<

^p

< .

^{Then forevery}

z

e E+,

the analyticWiener

tel I:(F; z)

^of

^F

^{isexpreed follows:}

z:(F; z) z xp{ ( z)

[(a,, ^{)]}. ()(). (.)}

PROOF. By Lemma 3.2,

the right hand sideof

(3.3)

^hasafinite value. Using Theorem

2.4,

weobtainthat

exp{ (1 z)

-[(h, x)~]} F(x)u(dx)

2

exp{ (1-z)

]

[(h,,l l.I((,,) (h )(

I()’exp{-ll ^}

and

d _du...du. Therefore,

we have where u

(u,,.-. ,u,)

6

R", Il E,=u,,

establhedtheequality

(3.3)

^by

(3.1)in

^Theorem3.1

Now

weshallexpress theanalytic

Feynman inteal I"t(F; q)

^of

F (n; 1)

^{the litof}

aquence ofWiener

inteals

^onthe

^atract

Wiener space.

THEOREM

3.4

Let (B,H,v)

beanabstract Wienerspace and let

{h,... ,h,}

^{be in}

Definition 2.3.

t F

6

(n; 1)

be givenby

(2.7).

^If

{z}=

^{is a}quenceof complexnumbers from

C+

^{such that}

z

^{approhes -iq} through

C+,

where q is a non-zero real number and

-1,thenthe

analic Feynman inteal I"(F; q)

^of

F

isexpressed follows:

i,(F;q) 2(z)

⁹

^{exp{ (1 z)} [(h, ^{x)]’} F(x)u(dx)} (3.4) PROOF. By

Theorem

2.4,

we canshow that

(z) exp{ [(h,, ^{x)]’} F(x) U()}

=1

Zk Zk

() f().exp{-Tll ^}d,

and

d _du...du.

Using the argument where u

(u,,.-. ,u,)

6

R

",

1[= =,u,

similar tothat theproofofTheorem

3.1,

weconcludethat

lim

<z)/ ^{exp{ 1- z)[(h,,}

₂₌₁

^{z).]} F(z)()}

2i()e I().exp{-ll ^}d

-iq iq

where the last equality follows from

(3.2)

ⁱⁿ Theorem 3.1 Thus the

proof

ofthistheorem is complete.

Now

we canobtaina

change

of scaleformula for Wiener integralson

(n; p)

which follows from Theorem 3.3and Definition2.2.

THEOREM

3.5

Let (B, H,)

^beanabstract Wiener space.

Let

p

>

^{0 begiven and let}

{hi,--., h,,}

^{beasin Definition}2.3. Then forevery

F

6

’(n;p),

/s F(px) u(dx) p-" fe ^{exp{ (p}

^2p

^l) [(h,,x)"12} F(x) ,(dx), (3.5)

78 Y.S. KIM where 1

PROOF.

First,^we canshow that for all real z

>

0,

I(F;z) =/B F(z-1/2x)v(dx)

by Definition2.2. Using Theorem 3.3 and taking z

p-2

in the aboveequality, we have the desiredresult.

EXAMPLE Let (B,H,v)

beanabstract Wiener

Space

and n beapositive integer and let

{hi,... ,h}

^beanorthonormal setof elements of

H.

Define

F B C

by

F() f((h,,)~,..., (h., ) ~) exp[--a ((h,, ^x)~) 21

3-’1

(3.6)

wherea isacomplex numberwith

Re(a) >

^0.

It

iseasy toseethat

F

E

f31<<o’(n p)

^since

Re(a) > 0,

andso

F

satisfiesthehypothesis ofall the theorems in this paper.

But

becauseofthespecial form of

F(see(3.6))

^{we can}easily evaluate the integralsonthe right-hand sideofequations

(3.1)

^and

(3.2). Thus,

fornon-zero real q and z E

C+,

^{it followsthat}

I(F" z) (--g) "

^{and that}

^{Il(F q)}

^lim

^{qI(F z) (__=z)}

^2--q

^r

ACKNOWLEDGEMENT

Workonthis paperwaspartlysupportedby GlobalAnalysis Research

Center,

SeoulNational University,

Korea.

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