A change of scale formula for Wiener integrals on the product abstract Wiener spaces

A CHANGE OF SCALE FORMULA FOR WIENER INTEGRALS ON THE PRODUCT ABSTRACT WIENER SPACES

YOUNG SIK KIM, JAE MOON AHN, KUN SOO CHANG AND

IL

1. Introduction

It has long been known that Wiener measure and Wiener measur- bility behave badly under the change of scale transformation [3] and under translation [2]. However, Cameron and Storvick [4] obtained the fact that the analytic Feynman integral was expressed as a limit of Wiener integrals for a rather larger class of functionals on a classical Wiener space. And then they found a rather nice change of scale for- mula for Wiener integrals on a classical Wiener space [5]. In [10,11,12], Yoo, Yoon and Skoug extended these results to Yeh-Wiener space and to an abstract Wiener space.

The purpose of this paper is to show the existence of the ana- lytic Feynman integral for certain functionals on the product abstract Wiener space and to obtain the relationships between the analytic Feynman integrals and the abstract Wiener integrals. And then using these results, we obtain change of scale formulas for abstract Wiener integrals on the product abstract Wiener space. Finally, we will show that most of theorems given in [10] become corollaries of our theorems.

2. Definitions and Preliminaries

Let H be a real separable infinite dimensional Hilbert space with inner product (.,.) and norm 1·1

..;r::). Let 11·110 be a measurable norm on H with respect to the Gaussiancylinder set measure 11 on H.

Received November 30, 1994.

1991 AMS Subject Classification: 28C20.

Key words: Abstract Wiener measure, Change of scale formula.

This paper was supported in part by GARC, Yonsei University Grant, BSRIP, Ministry of Education and KOSEF in 1995.

Let B denote the completion of H with respect to 11·/10. Then it is well known [9] that B becomes a separable Banach space. Let i denote the natural injection from H into B. The adjoint operator i* is one-to-one and maps B* continuously onto a dense subset of H*. By identifying H* with H and B* with i*B*, we have a triplet (B*, H, B) in such a way that B* c H* == H c B and (y,x)

(y,x) for all y

B* and x E H, where (.,.) denotes the natural dual pairing between B* and B. By a well-known result of Gross, /-l

has a unique count ably additive extension m to the Borel O"-algebra B(B) on B. Then we say that the triplet (B, H, m) is an abstract Wiener space and m is an abstract Wiener measure. For more details, see [9].

Let F be a complex-valued measurable functional on (B, H, m).

Then we will denote the abstract Wiener integral of F with respect to m by

l F(x) dm(x).

Let (en) denote a complete orthonormal system on H such that en's

are in B*. For each h E H and x E B, define a stochastic inner product (., .)"" between H and B as follows :

(2.1) "" {lim ^{t (h,} ek)( ek, x), (h, x)

0,

if the limit exists otherwise

It

is well known [8] that for every h

H, (h,

exists for m - a.e x

B, and is a Borel measurable function on B having a Gaussian dis- tribution with mean zero and variance Ih1

Also if both h and x are in H, then (h,x)""

(h,x). Furthermore, it is easy to show that for each a E lR,(ah,x)"" = a(h,x)"" = (h,ax)"", x E B, hold and that (h, x)"" = (h, x) m-a.e. x E B if h E B*.

is well known [8]

that (h, x) "" is essentially independent of the choice of the complete orthonormal system used in its definition.

DEFINITION

2.1. Let B

xj=IB denote the product of v copies

of B and let F be a complex-valued measurable functional on B

such

that the integral

exists for all Ak > 0,

= 1,2,'" ,v, where rn" is the product abstract Wiener measure on B", X = (A},'" ,A,,)

JR" and x = (x},," ,x,,)

B". If there exists an analytic function J*(F; z) on

on

= {z = (Zl,' .. ,z,,) E Cv : Re(zk) > ° ^for

^{= 1,2,'" v} such}

that J*(F; X) = J(F; X) for all Ak > 0,

= 1,2",', v, then we define J*(F; Z) to be the analytic Wiener integral of F over B" with parameter z, and for zEn we write

(2.3) rW(F; Z) = ^J*(F; Z).

Let if= (ql,'" ,q,,) E JR" be such that qk =I- ° ^{for k =} 1,2"" ,v. ^If

the following limit (2.4) exists, we call it the analytic Feynman integral of F over B" with parameter if and we write

(2.4) rf(F'

=

lim Jaw(F-.z"I

where z = (z}, ... ,z,,) approaches -iif = ( -iq}, ... ,-iq,,) through n.

Given two complex-valued functionals F and G on

we say that F = G s-a.e. if for all ak > 0, k =

,v, F(alxl,'" ,a"x,,)

= G( alx}"" ,a"x,,) for rn"-a.e. x ^{in B".} For a functional F on B"

, we will denote by [F] the equivalence class of functionals which are equal to F s-a.e.

DEFINITION

2.2. Let F(B") be a class of all equivalence classes of functionals on B

which have the form

for some finite complex Borel measure

on H. In particular, when v = 1, F(B") is reduced to F(B), which is the Fresnel class on the abstract Wiener apace (B, H, rn).

Let M(H) denote the space of all finite complex Borel measures

on H. Then it is well known that M(H) is a Banach algebra under

convolution, with the norm 1IJ.l11 equal to the total variation of

E

M(H). By using Propositions 3.2 or 5.1 in [8], we can prove that the

mapping

[F] is a Banach algebra isomorphism where

and F

(3.1)

are related by (2.5) and hence :F(BlI) becomes a Banach algebra under the norm II ^F II

IIJLII.

The following theorem, which is quoted from [8], is necessary for proving the existence of the analytic Feynman integral for the func- tionlas F belonging to :F( B lI ).

THEOREM

2.1. Let D be an open subset ofC

where C

Xj=1C is the product ofk copies ofthe complex plane C. Assume that 9 : D

C

continuous and analytic in each variable separately. That is, for each j,

:s ^j :s ^k, and each point (Zb'" ,Zj-b ZjH ... ,Zk)

C k- 1 such that Dj

{Zj E C: (Zl,'" ,Zj,'" ,Zk) E D} is nonempty, the function j(Zj)

g(Zl, ... ,zj,'" ,Zk) is analytic in Dj. Then 9 is analytic as a function of k complex variables in D.

D is connected and contains the set D+

{(Zl,'" ,Zk) ECk: Re(zj) > 0,1 :s

:s

k}, then 9 is uniquely determined by its restriction to D+.

3. Change of Scale Formulas

We begin this section with obtaining the existence theorem of the analytic Wiener and Feynman integrals for the functionals F given by (2.5).

Throughout this section, we will let

1\=

{X =

,All)

RlI :

> 0,1:Sj:S 1I}, and

n

{z= (Zl,'" ,ZlI) E C lI : Re(zj) > 0, 1 :s ^j :s ^1I}.

THEOREM

3.1. Let FE :F(BlI) be given by (2.5). Then the analytic Wiener integral of F over BlI with parameter z

(Zl,'" ,ZlI)

n

exists, and

rW(F; Z)

_lH I exp{-~ _2. t ^~lhI2}

^dJL(h).

)=1

Also the analytic Feynman integral of F over B lI with parameter if

(qb'" ,qll) exists, provided that qk

° ^{for all k}

1,2"" ,1I, and

Proof. Using Fubini's theorem, we have

J(F}.) = f f exp{i t(h, Aj~ Xj)"'} dp,(h)dmV(x)

lBv lH ^j=l

= f f ^{eXP{i t(h,} Aj~ Xj)"'}dmV(x) dp,(h)

lH lBv ^j=l

= L ^exp{ -~ t. ^:j [hi' } dl'(h)

for all X = (Ab'" ,Av) E

where the last equality is established from the fact that each (h, x j)'" is Gaussian random variable with mean zero and variance Ihl

with respect to the abstract Wiener measure m.

Define

J*(F; Z) = _llH f exp{-~

^t ~lhI2}

^dp,(h)

]=1

for all z

no = {z = (Zl"" ,zv)

Cv : Zj

0, Re(zj)

0, 1

j S;

v}. Then r(F;·) exists for all z

no, and is continuous on no

by the dominated convergence theorem. Since J*(F; X) = J(F; X) for all X

it is enough to show that the restriction of r(F;·) to n is an analytic function in order to show that equations (3.1) and (3.2) are established. Using Morera's theorem, we can show that J*(F;·) is analytic in each complex variable separately in n. With the assistance of Theorem 2.1, we can conclude that the restriction of J*(F; .) to n

is an analytic function.

The following lemma plays a key role for obtaining relationships be- tween the abstract Wiener integral and the analytic Feynmam integral on an abstract Wiener space (B, H, m), which is quoted from [10].

LEMMA 3.2. Let Z

C with Re(z) > 0, let {ej:

=

,n} be an orthonormal set in H, and let h EH. Then

(3.3) f exp{1;zt[(e j ,x)",]2+ i (h,X)"'}dm(X)

lB

" _{Z-l~[ ]2

2}

=z-"2exp - L - J (ej,h) -2 1hl .

2z .

]=1

In the following two theorems 3.3 and 3.5, for every F E F(BV), we express the analytic Wiener integral and the analytic Feynman integral of F over BV as a limit of a sequence of abstract Wiener integrals, respectively.

THEOREM

3.3. Let (en) be a complete orlhonormal system in H and let z

(ZI'··· ,zv)

Let

F(BV) be given by (2.5). Then we have

(3.4)

rW(F;Z)

n~CXl(fI Zj)~ lv ^{exp{t 1} ~Zj t[(e k,Xj)''']2}

]=1 B ]=1 k=1

. F(x) dmV(x),

v

where dmV(x) = dm(xl) dm(x2)'" dm(x v), and n ^Zj = ZI . Z2··· Zv·

j=1

Proof. Since F E F(BV) is given by (2.5),

F(x) = 1 exp{it(h,Xj)'''}d/L(h), xE ^{B V ,}

H j=1

for some /L

M(H). By Fubini's theorem and Lemma 3.2, we obtain

1.. ^{exp{~ 1 ~}

^{t. [(e.,}

}F(X)dm"(X)

=

(fIZj)-~ [exp{tZ~~.lt[(ek,h)]2}exp{-ilhI2}d/L(h).

]=1

lH

Next, using the bounded convergence theorem, the equation (3.1) and Parseval's relation, it follows that

n~(Q

r ^{1.. exp{ ~ 1 ~}

t.[(e.,Xj)-j'} F(X) dm"(X)

= 1. ^exp{ -~ ~ ~ ^{Ihl'} dl'(h)}

=

rW(F;Z).

COROLLARY

3.4. Let (en) be a complete orthonormal system in H and let F

F( B) be given by

F(x) = L ^exp{i(h, x)"'} dJ.L(h), x E B

for some J.L E M(H). Let Z E C with Re(z) > O. Then the analytic Wiener integral of F over B with parameter z exists, and

(3.5)

• n {

{1 - ^{z ~ '"} 2}

rW(F; z) = n~~ z2" lB ^{exp - 2 -} ~ [(ek, x)] F(x)dm(x).

Proof.

Apply Theorem 3.3 after making the following choices:

1/=

1,

z.

REMARK

1. The first part of Cororally 3.4 coincides with that of Proposition2.2 in [8], and the equation(3.5) of Cororally 3.4 coincides with the equation(3.3) of Theorem 3 in [10] by taking

= (z) therein.

THEOREM

3.5. Let (en) be a complete orthonormal system in H and let F E F(B

be given by (2.5). Let (Zk,n) be a sequence of complex numbers such that Re(zk,n) > 0 and Zk,n

-iqk(qk

0) as n

for

= 1,2,···

Tben

(3.6)

rf(F; if)

= n~oo[iI ^Zj,n] -¥-1" ^{exp{t 1-} ₂ ^{Z j ,n t[(e k ,x}

^)"']2}

J=l B J=l

k=l

. F(x) dmll(x), wbere

IT

^{Zj,n =} Zl,n Z2,n ... zlI,n, and j=l

dmll(x) = dm(xl)dm(x2)·· ·dm(xll ).

Proof. To prove this theorem, we modify the proof of Theorem 3.3 by replacing "z/' by "Zj,n" wherever it occurs, and by replacing "-2z/' by "2q

i" in the last equality in the proof of Theorem 3.3.

COROLLARY

3.6. Let (en) be a complete orthononnal system in H and let F

:F(B) be given by

F(x)

L ^exp{ i(h, x)-} d/-L(h), x

B,

for some /-L

M(H). Let (zn) be a sequence of complex numbers such that Re(zn) > 0 for

n

N, and Zn

-iq(q =I 0) as n

Then the analytic Feynman integral of F over B with parameter

exists, and

(3.7)

raf(F;q)

!im zj ( ex p {l-Z

~[(ek,x)' ..]2}F(x)dm(x).

n-oo

JB ^{2 L...J}

Proof. Apply Theorem 3.5 after making the following choices; v

1, Zl,n = Zn, and ql = q.

REMARK

2. The:first part of Corollary 3.6 coincides with the second part of Proposition 2.2 in [8], and the equation(3.7) of Corollary 3.6 coincides with the equation(3.2) of Theorem 2 in [10].

Now, we can obtain our main result, namely a change of scale for- mula for abstract Wiener integrals on a product abstract Wiener space by using Theorem 3.3.

THEOREM

3.7. Let Pk > 0, k = 1,2,,,, ,v be given and let (en) be a complete orthononnal system in H. Let F

:F( BV) be given by (2.5). Then we have

(3.8)

( F(PIXll"', pvxv)dmV(x)

JBv

[ V]

{V

1

= n~oo IIpj ^{Jp exp L} P~ ~ L[(ek,Xj)-]2 F(x)dmV(x),

j=l B j=l

Pl

(3.9)

v

where Il Pj

PIP2'" Pv and dmV(x)

dm(xI)dm(x2)'" dm(x v).

j=1

Proof. Replacing Zj by pj-2 in the equation (3.4), we have

r F(PIXb"', pvxv)dmV(x)

JBP

~ n~~~J}}r fa. exp{t, p~:? ¹ t,[(e" ^{x;)-J' }} F(x)dm"(X) COROLLARY 3.8. Let P > 0 and let (en) be a complete orthonormal system in H. Let F be a functional defined by

F(x)

i eXP{i(h, x)""'} dJ.t(h), x E B, for some J.t

M (H). Then we have

L ^F(px)dm(x)

= lim p-

r ^exp{P: ~1 ^t[(e k,x)"",]2}F(X)dm(x).

n--oo

J

^P

Proof. Apply Theorem 3.7 after making the following choices:

v = 1, PI =

REMARK 3. Corollary 3.8 coincides with Theorem 4 in [10]

The Banach algebra :F(BV) is not closed with respect to pointwise or even uniform convergence [7,p.2], and its uniform closure Clu(:F(BV)) with respect to uniform convergence

-a.e. is a larger space than :F(BV). Now we will show that the equation (3.8) also holds for every F

Clu(:F(BV)).

THEORM 3.9. Let Pk > 0, k = 1,2, ... ,v be given and let (en) be a complete orthonormal system in H. Then the equation (3.8) holds for every F E Clu(:F(BV)).

Proof. Since F E Clu(:F(BV)), there exists a sequence (Fp) from :F(BV) such that F(x) = !im Fp(x) uniformly s -a.e. on BV. Also

p--oo

since each Fp

:F(B"'), Fp(x) exists and is bounded s-a.e on B'" for every pEN. From the definition of uniform converence s-a.e., it follows that for Pie> 0, k = 1,2, ... ,1/,

(3.10)

uniformly a.e. on B"', and (3.11)

[ F(PIXll···, p",x",)dm"'(x) = lim [ Fp(PIXll···, p",x",)dm"'(x).

Jw p-=Jw

Now taking z =

and h = 0 in the equation (3.3) of Lemma 3.2, we obtain

(3.12)

Since F(x) = lim Fp(x) uniformly s-a.e. on B"', there exist M > 0

p-oo

and a scale-invariant null set N of B'" such that for all pEN and all

xE ^B"'-N,

(3.13) IFp(x) I

M and IF(x)1

M.

Hence, using (3.12) and (3.13), we obtain

~2M.

Finally, using Theorem 3.7, the iterated limit theorem and the dom- inated convergence theorem, we have

f F(P1 Xll"" PII XII) dm

(x)

lB"

=

lim f Fp(P1Xll"" PII XII) dm

(x)

p ...

oolB"

[ 11 ]

1 ^{1I

¹

= lim lim IIpj exp ~

~ ~[(ek,Xjr'J2

p ...oo n ... oo B" P "

j=1 j=1 )

k=1

· Fp(x) dm

(x)

[ 11 ]

1 ^{1I

¹

=

lim lim IIpj exp ~ P~ ~ ~[(ek,Xj)"'J2

n ...oop ... oo

j=1

j=1 P

k=1

· Fp(x) dm

(x)

[

11 ] -n

f {1I

1

= n~ !! ^{Pi lB" exp} t; P~~ {;[(ek'Xi)~]2

· F(x) dm

(x).

Thus the equation (3.8) holds for every F E Clu(:F(BII))

Taking v = 1 and P1 = P in Theorem 3.9, we have the following corollary which cioncides with Theorem 5 in [10]:

COROLLARY

3.10. Let P > 0 be given and let (en) be a complete orthonormaJ system in H. Then for every FE Clu(:F(B))

1 ^{F(px )dm( x )}

=

lim

p-

lB f exp{p2~1~[(ek,X)""'J2}F(X)dm(X),

k=1

where :F(B) is the Fresnel class on B.

Finally, we shall explicitly compute an abstract Wiener integral of certain functional and then check that the equation (3.8) is established.

Consider a functional F defined by

(3.14) F(x) = exp{a t(h,Xj)"'},x = (Xl,'" ,XII) E B II ,

)=1

fork=n+1.

for k = 1,2, ... ,n.

where h is any fixed element of H and a is a real or complex number.

By the abstract Wiener integration formula, we have (3.15)

f ^{F(PI X}

PIIXII) dm"(x) = f exp{a t pj(h, Xj)"'} dm"(x)

~v J~ ~1

~ 11 ^{fa exp{apj(h,xj)-} dm(Xj)} = expUa21h1' ~ 4

where

{

(ek,h),

Ck = [Ihl 2 - ~1 ^{(ek' h)2]} ~,

The right hand side of (3.16) is equal to the followings :

(3.17)

Hence

-[ll, ^Pi r ^{exp{ ~ t. P~ t, (ek' h)' }}

exp{ 1/;2 ^{(l h l 2 -} t,(e

^{h)2) }.}

Thus we have shown that the equation(3.8) is satisfied for the func- tional F given in (3.14). In particular, if a is pure imaginary, then F belongs to :F(BIJ). On the other hand, if Re(a) "I 0, then F is un- bounded, so F rt :F(B IJ ) and also F rt Clu(:F(B IJ )). Thus this example shows that the class of functionals for which the equation (3.8) holds is more extensive than Clu(:F(BIJ)).

References

1. J. M. Ahn, G. W. Johnson and D.1. Skoug, Functions in the Fresnel class of an Abstract Wiener Space, J. Korean Math. Soc. 28 (1991), 245-265.

2. R. H. Cameron, The Translation Pathology of Wiener Space, Duke Math J. 21 (1954), 623-628.

3. R. H. Cameron and W. T. Martin, The Behavior of Measure and Measurability under Change of Scale in Wiener Space, Bull. Amer. Math. Soc. 53 (1947), 130-137.

4. R. H. Cameron and D. A. Storvick, Relationships between the Wiener Integral and the Analytic Feynman Integral, Supplemento ai Rendiconti del Circolo Matematico di Palermo Serie II-Numero 17 (1987), 117-133.

5. , Change of Scale Formulas for Wiener Integral, Supplemento ai Rendi- conti del Circolo MatematicodiPalermo Serie II-Numero 17 (1987), 105-115.

6. D. M. Chung, Scale-Invariant Measurability m Abstract Wiener Space, Pacific J. Math. 130 (1987),27-40.

7. G. W. Johnson and D. L. Skoug,Stability Theorems for the Feynman Integral, Supplemento ai Rendiconti del Circolo MatematicodiPalermo Serie II-Numero 8 (1985), 361-367.

8. G. Kallianpur and C. BromIey, Generalized Feynman Integrals using Analytic Continuation in Several Complex Variables, In Stochastic Analysis and Appli- cations {ed.M.H.Pinsky}, Dekker, New York, 1984, pp. 217-267.

9. H. -H. Kuo, Gaussian Measures in Banach Spaces, vol. 463, Springer Lecture Notesin Math, Berlin, 1975.

10.I. Yoo and D. L. Skoug, A Change of Scale Formula for Wiener Integrals on Abstract Wiener Spaces, Internat J. Math and Math. Sci. 17 (1994), 239-248.

11. ,A Change of Scale Formula for Wiener Integrals on Abstract Wiener Space Il, J. Korean Math. Soc. 31 (1994), 115-129.

12.I. Yoo and G. J. Yoon, Change of Scale Formulas for Yeh-Wiener Integrals, Comm. Korean Math. Soc. 6 (1991), 19-26.

Young Sik Kim

Global Analysis Research Center Department of Mathematics Seoul Notional University Seoul 151-742, Korea Jae Moon Ahn

Department of Mathematics Education Kon-kuk University

Seoul 133-701, Korea Kun Soo Chang

Department of Mathematics Yonsei University

Seoul 120-749, Korea

n Yoo

Department of Mathematics Yonsei University

Kangwondo 227-701, Korea