A Change of Scale Formula for Wiener Integrals about the First Variation on the Product Abstract Wiener Space

Article

A Change of Scale Formula for Wiener Integrals about the First Variation on the Product Abstract Wiener Space

Young Sik Kim

Citation:Kim, Y.S. A Change of Scale Formula for Wiener Integrals about the First Variation on the Product Abstract Wiener Space.Symmetry2021,13, 12.

https://dx.doi.org/10.3390/sym13010 012

Received: 30 November 2020 Accepted: 17 December 2020 Published: 23 December 2020

Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional claims in published maps and institutional affiliations.

Copyright:© 2020 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/

licenses/by/4.0/).

Department of Mathematics, College of Natural Sciences, Industry-University Cooperation Foundation, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Korea; [email protected]

Abstract:We shall prove the existence of the Wiener integral and the analytic Wiener and Feynman integral and we obtain those relationships and later, we prove the change of scale formula for the Wiener integral about the first variation of a function defined on the product abstract Wiener space.

Later, we obtain those relationships in the Fresnel class as it’s corollaries.

Keywords:abstract Wiener space; Fourier–Feynman transform; change of scale formula

1. Introduction

It has been known that Wiener measure and Wiener measurability behave badly under the change of scale transformation [1] and under translation [2]. However, Cameron, R.H.

and Storvick, D.A. [3] proved that an analytic Feynman integral was expressed as a limit of Wiener integrals for a rather larger class of functionals on the Wiener space. They found a nice change of scale formula for Wiener integrals on the Wiener space [4]. Yoo, I. and Skoug, D. and Yoon, G. J. extended those results to an abstract Wiener space and Yeh–Wiener space in [5,6].

In [2,7,8], Cameron, R.H. and Martin, W.T. investigated the behavior of Wiener inte- grals on transformations and translations in 1945 and in 1947. In [1], they developed the behavior of measures and measurability under the change of scale on the Wiener space in 1947. In [9], Cameron, R.H. and Storvick, D.A. introduced a Banach algebraSof func- tionals onC₀[0,T]. In [10–12], Kim, Y.S. proved the change of scale formula for the Wiener integral on the abstract Wiener space about the cylinder functions and the unbounded cylinder function and proved the relationship between the analytic Feynman integral and the Wiener integral and the first variation. In [13], Kim, Ahn, Chang, and Yoo prove the change of scale formula for the Wiener integral on the product abstract Wiener space in the classF(B^ν).

In this paper, we prove the existence of the analytic Feynman integral and prove the change of scale formula about the first variation for functions inF(B^ν)of the form : G(~x) = R

Hexp{i ∑^ν

j=1

(h,x_j)^∼}dµ(h),µ∈ M(H)on the product abstract Wiener space.

We show that the first variation of the analytic Wiener integral and the analytic Feynman integral ofG(~x)can be perfectly expressed as the limit of the sequence of Wiener integrals of the first variation. Finally, we prove that for ν = 1, a change of scale formula for the Wiener integral about the first variation ofF(x) in the Fresnel classF(B) holds as its corollaries.

2. Definitions and Preliminaries

LetHbe a real separable infinite dimensional Hilbert space with inner producth·,·i and norm| · |=^ph·,·i. Let|| · ||₀be a measurable norm onHwith respect to the Gauss measureµ. Let B denote the completion of H with respect to|| · ||₀. Let idenote the natural injection fromHintoB. The adjoint operatori^∗ ofiis one-to-one and mapsB^∗ continuously onto a dense subset ofH^∗, whereH^∗andB^∗are topological duals ofHandB,

Symmetry2021,13, 12. https://dx.doi.org/10.3390/sym13010012 https://www.mdpi.com/journal/symmetry

respectively. By identifyingHwithH^∗andB^∗withi^∗B^∗, we have a triplet(B^∗,H,B)such thatB^∗ ⊂H^∗≡ H⊂Bandhh,xi= (h,x)for allhinB^∗andxinH, where(·,·)denotes the natural dual pairing betweenB^∗andB. By a well-known result of Gross [14],µ·i⁻¹ has a unique countably additive extensionmto the Borelσ-algebraB(B)onB. The triplet (H,B,m)is called an abstract Wiener space andmis called a Wiener measure. We denote the Wiener integral of a functionalFbyR

BF(x)dm(x). For more details, see [15–17]. Let {ej}^∞_j=1denote a complete orthonormal system inHsuch thatejs are inB^∗. For eachh∈ H andx ∈B, we define a stochastic inner product(·,·)^∼betweenHandBas follows:

(h,x)^∼ =







limn→∞

∑n j=1

hh,eji(ej,x), if the limit exists,

0 otherwise.

(1)

It is known in [16] that for everyh ∈ H,(h,x)^∼ exists form−a.e.x ∈ Bin and it has a Gaussian distribution with mean zero and variance|h|². Furthermore, it is easy to show that (h,x)^∼=hh,xiform−a.e.xinBifh∈B^∗,(h,x)^∼is essentially independent of the choice of the complete orthonormal system used in its definition, and finally that if{h₁,· · ·,h_k}is an orthonormal set of elements inH, then(h₁,x)^∼,· · ·,(h_k,x)^∼are independent Gaussian functionals with mean zero and variance one. Note that if bothh andxare inH, then (h,x)^∼=hh,xi.

Letνbe a positive integer and letm^νdenoteν-dimensional abstract Wiener measure and letB^ν = B×B× · · · ×B(νtimes). We denote the Wiener integral of a function F defined onB^νbyR

B^νF(~x)dm^ν(~x). A subsetEofB^νis said to be scale-invariant measurable ifρEis Wiener measurable for eachρ>0 and a scale-invariant measurable setNis scale- invariant null providedm^ν(ρN) =0 for eachρ > 0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s.a.e.). See [18]

for more about the scale-invariant measurability in abstract Wiener space.

Throughout this paper, letRⁿ denote the n-dimensional Euclidean space and let C, C+, andC^∼₊ denote the complex numbers, the complex numbers with positive real part, and the nonzero complex numbers with nonnegative real part, respectively. Let Γ0={(z1,z2,· · ·,zν)∈C^ν:zj6=0,Re(zj)≥0, 1≤j≤ν}and letΓ={(z1,z2,· · ·,zν)∈ C^ν:zj6=0,Re(zj)>0, 1≤ j≤ν}.

Definition 1. Let G be a complex-valued measurable function on B^νsuch that the integral

J(G;~_λ) = Z

B^νG(λ⁻

1 2

1 x1,λ⁻

1

2 2x2,· · ·,λ⁻

1 2

ν xν)dm^ν(~x) (2) exists for all~_{λ, with}~_λ= (λ₁,λ₂,· · ·,λ_ν) ∈ R^ν,λ_j > 0, 1≤ j≤ ν. If there exists a function J^∗(G;~z)analytic onΩ such that J^∗(G;~_λ) = J(G;~_λ)for all~_{λ, with}~_λ = (λ₁,λ₂,· · ·,λ_ν) ∈ R^ν,λ_j>0, 1≤j≤ν, then we define J^∗(G;~z)to be the analytic Wiener integral of G over B^νwith parameter~z, and for each~z∈Ω, we write

Z anw_~z

B^ν G(~x)dm^ν(~x) = J^∗(G;~z). (3)

Let~q= (q1,q2,· · ·,qν)∈R^ν, where qj,1≤j≤ν, is a nonzero real number and let G be a function on B^νwhose analytic Wiener integral exists for each~z inΓ. If the following limit exists, then we call it the analytic Feynman integral of G over B^νwith parameter~q, and we write

Z _{an f~q}

B^ν G(~x)dm^ν(~x) = lim

~z→−i~q Z _anw~z

B^ν G(~x)dm^ν(~x), (4)

where~z approaches−i~q throughΓand i²=−1.

LetM(H)be the space of complex-valued countably additive measures defined on B(H), the Borel class ofH. A complex-valued countably additive measureµnecessarily has finite total variation||µ||. Under the norm|| · ||and with convolution as multiplication, M(H)is a commutative Banach algebra with identity.

Now, we will introduceF(B^ν)in [13].

Definition 2. The classF(B^ν)is defined as the class of functions G:B^ν→C defined on B^νof the form

G(~x) = Z

Hexp

i

∑

ν j=1

(h,x_j)^∼

dµ(h) , (5)

where~_x= (x1,x2,· · ·,xν)∈B^ν, µ∈ M(H).

Remark 1. Forν =1,F(B)is a Fresnel class of functions F(x) =R

Hexp

i(h,x)^∼

dµ(h), which is a stochastic Fourier transform of a measureµ∈ M(H).

In [19], R.H. Cameron introduced the first variation theory of an indefinite Wiener integral on the Wiener space. We define the first variation onF(B^ν).

Definition 3. Let G be a function onF(B^ν). Then, for~u∈ B^ν, the function defined by δG(~x|~u) = ^∂

∂hG(~x+h~u)|_h=0 (6) is called the first variation of G in the direction~u (if it exists).

The next theorem, which is quoted from [14], is necessary for proving theorems in F(B):

Theorem 1. Let U be an open subset of C^k, where C^K = ×^k_j=1C is the product of k-copies of the complex plane C. Assume that f : U → C is continuous and analytic in each variable separately. That is, for each j, 1 ≤ j ≤k, and for each point(z1,z2,· · ·,zj−1,zj+1,· · ·,z_k)∈ C^k−1such that U_j={z_j ∈C:(z₁,z2,· · ·,z_j,· · ·,z_k)∈U}is non-empty, the function h(z_j) = f(z1,z2,· · ·,z_j,· · ·,z_k)is analytic in U. Then h is analytic as a function of k complex variables in U. If U is connected and contains the set U⁺={(z₁,· · ·,z_k)∈C^k:Re(z_j)>0, 1≤j≤k}, then h is uniquely determined by its restriction to U⁺.

In the next section, we will use the following well-known integration formula:

Z

Rexp

−cu²+idu

du= rπ

c exp

−^d

2

4c

, (7)

wherecis a complex number withRe(c)>0,dis a real number, andi²=−1.

3. Main Results

LetG∈ F(B^ν)be of the form(5). Let~u= (u1,u2,· · ·,u_ν)withu_j∈ Hand|u_j|<_∞, 1≤j≤ν. Assume thatR

H |h|d|µ|(h) < ∞. Lemma 1.

δG(~x|~u) = Z

H

i

∑

ν j=1

(h,u_j)^∼

exp

i

∑

ν j=1

(h,x_j)^∼

dµ(h). (8)

Proof. By Definition3of the first variation, δG(~x|~u)

= _∂l^∂ G(~x+l~u)|_l=0

= _∂l^∂ R

H exp

i ∑^ν

j=1

(h,xj+l uj)^∼

dµ(h)|_l=0

= R

H ∂

∂lexp

i ∑^ν

j=1

(_h,xj+_{l uj})^∼

dµ(_h)|_l=0

= R

H

i ∑^ν

j=1

(h,u_j)^∼

exp

i ∑^ν

j=1

(h,x_j)^∼

dµ(h).

(9)

Foru_j∈ H,(h,u_j)^∼=hh,u_jiand|hh,u_ji|<|h| · |u_j|for 1≤j≤ν. Therefore,

|δG(~x|~u)|

=

R

H

i ∑^ν

j=1

(h,uj)^∼

exp{i ∑^ν

j=1

(h,xj)^∼

dµ(h)

≤ _∑^ν

j=1

R

H

∑ν j=1

(h,u_j)^∼d|µ|(h)

≤ _ν

j=1∑

|u_j|

R

H|h|d|µ|(h)<_∞.

(10)

And soδG(~x|~u)exists.

We can deduce the following result inF(B)on the abstract Wiener space(H,B,m): Corollary 1. For G∈ F(B)and for s.a.e.u∈ B,

δG(x|u) = Z

H

i(h,u)^∼

exp

i(h,x)^∼

dµ(h) (11)

Now, we have the analytic Wiener integral and the analytic Feynman integral for the first variation of functions inF(B^ν)_.

Theorem 2. (1). For every~z= (z₁,z2,· · ·,z_ν)∈_Ω0, Z _anw~z

B^ν δG(~x|~u)dm^ν(~x) = Z

H

i

∑

ν j=1

(h,uj)^∼

exp

− |h|²

∑

ν j=1

1 2zj

dµ(h), whereΩ0={(z₁,z₂,· · ·,z_ν)∈C^ν:z_j 6=0 ,Re(z_j)≥0 , 1≤j≤ν}.

(2).

Z an f_~q

B^ν δG(~x|~u)dm^ν(~x) = Z

H

i

∑

ν j=1

(h,u_j)^∼

exp

− |h|²

∑

ν j=1

i 2q_j

dµ(h), (12) {z_j,k}_k → −i qjthroughΩfor each j=1, 2,· · ·,ν.

Proof. For~_λ= (λ₁,λ₁,· · ·,λ_ν)∈R^νwithλ_j>0 , 1≤j≤ν,

Ranw~_λ

B^ν δG(~x|~u)dm^ν(~x)

= R

B^νδG(λ₁⁻

1

2x1+y1,· · ·,λ_ν⁻

1

2xν|~u)dm^ν(~x)

= R

B^ν

R

H

i ∑^ν

j=1

(h,uj)^∼

exp

i ∑^ν

j=1

(h,λ_j⁻¹²xj)^∼

dµ(h)

dm^ν(~x)

= R

H

i ∑^ν

j=1

(h,u_j)^∼ R

B^νexp

i ∑^ν

j=1

(h,λ_j⁻

1 2x_j)^∼

dm^ν(~x)

dµ(h)

= R

H

i ∑^ν

j=1

(h,u_j)^∼

exp

− |h|² _∑^ν

j=1 1 2λ_j

dµ(h).

(13)

By analytic continuation in~z= (z₁,z2,· · ·,z_ν)∈ _Ω0toΩ={(z₁,z2,· · ·,z_ν) ∈C^ν: z_j 6=0,Re(z_j)>0, 1≤j≤ν}, we obtain(12).

Now let{z_j,k}_kbe the sequence inΩ, with{z_j,k}_k → −i qjwheneverk→∞for each j=1, 2,· · ·,ν. Then

Ran f~q

B^ν δG(~x|~u)dm^ν(~x)

= limk→∞R^anw~_zk

B^ν δG(~x|~u)dm^ν(~x)

= limk→∞R

H

i ∑^ν

j=1

(h,u_j)^∼

exp

− |h|² _∑^ν

j=1 1 2z_j,k

dµ(h)

= R

H

i ∑^ν

j=1

(h,u_j)^∼

exp

− |h|² _∑^ν

j=1 i 2q_j

dµ(h),

(14)

whenever{z_j,k}_k→ −i q_jthroughΩfor eachj=1, 2,· · ·,ν.

We can deduce the following result inF(B)on the abstract Wiener space(H,B,m): Corollary 2. (1). For G∈ F(B)and for z∈C⁺and for s.a.e.u∈ B,

Z _anwz

B δG(x|u)dm(x) = Z

H

i(h,u)^∼

exp

− ¹ 2z|h|²

dµ(h), (15) where C⁺={z∈C : Re(z)>0}.

(2).

Z an f_q

B δG(x|u)dm(x) = Z

H

i(h,u)^∼

exp

− ⁱ 2q|h|²

dµ(h) (16) To expand the main result, let{e_j},j=1, 2,· · ·,n, be an orthonormal set in H and let h∈ H.

We need the following Lemma in [5] to prove properties in the next set of results.

Lemma 2([5]). Let z∈C with Re(z)>0. Then, R

Bexp

1−z 2

∑n j=1

(e_j,x)²+i(h,x)^∼

dm(x)

=z⁻ⁿ²exp

z−1 2z

∑n j=1

[hej,hi]²−¹₂|h|²

.

(17)

Theorem 3. For every~z∈Ω, exp

_ν

j=1

∑

(₁−z_j) 2

∑

n k=1

(e_k,x_j)²

δG(~x|~u)

is a Wiener integrable function of~x∈B^νand

R

B^νexp ν

j∑=1

(1−zj) 2

∑n k=1

(e_k,x_j)²

δG(~x|~u)dm^ν(~x)

= ∏^νj=1(zj)⁻ⁿ2R

H

i∑^ν

j=1

(h,uj)^∼

exp _ν

j∑=1

z_j−1 2z_j

∑n k=1

|he_k,hi|²

−^ν₂|h|²

dµ(h).

(18)

Proof. By (17),

R

B^νexp _ν

j=1∑ (1−z_j)

2

∑n k=1

(e_k,x_j)²

δG(~x|~u)dm^ν(~x)

= R

H

R

B^ν

i ∑^ν

j=1

(h,uj)^∼

exp _ν

j=1∑

[^(1−z₂ ^j) _∑ⁿ

k=1

(e_k,xj)²+i(h,xj)^∼]

dm^ν(~x)

dµ(h)

= _∏^ν_j=1(_zj)⁻ⁿ² R

H

i∑^ν_j=1(_h,uj)^∼

exp _ν

∑

j=1

z_j−1 2z_j

∑n k=1

|hek,hi|²

−^ν₂|h|²

dµ(_h)_.

(19)

The last formula in (19) has a finite value for~z∈Ω.

Corollary 3. For G∈ F(B)and for s.a.e.u∈ B, exp{¹−z

2

∑

n j=1

[(ej,x)^∼]²}δG(x|u) is a Wiener integrable function of x∈B, and

R

Bexp

1−z 2

∑n j=1

[(ej,x)^∼]²

δG(x|u)dm(x)

= z⁻ⁿ² R

H

i(h,u)^∼

exp

z−1 2z

∑n k=1

|he_k,hi|²

−¹₂|h|²

dµ(h).

(20)

Now, we prove that the analytic Wiener integral of the first variation inF(B^ν)can be perfectly expressed as the limit of Wiener integrals on(_H,B^ν_,m)_.

Theorem 4. For every~z∈_Ω, Ranw~z

B^ν δG(~x|~u)dm^ν(~x)

= limn→∞

∏^ν_j=1zj

ⁿ₂ R

B^νexp _ν

∑

j=1 (1−z_j)

2

∑n k=1

(ek,xj)²

δG(~x|~u)dm^ν(~x). (21) Proof. By the bounded convergence theorem and Parseval’s relation, we have

limn→∞

∏^νj=1z_j ⁿ₂

R

B^νexp _ν

j=1∑

(1−z_j) 2

∑n k=1

(e_k,x_j)²

δG(~x|~u)dm^ν(~x)

= limn→∞R

H

i ∑^ν

j=1

(h,uj)^∼

exp _ν

j=1∑ (z_j−1)

2z_j

∑n k=1

|he_k,hi|²

−^ν₂|h|²

dµ(h)

= R

H

i ∑^ν

j=1

(h,uj)^∼

exp _ν

∑

j=1

(z_j−1) 2z_j

∑∞ k=1

|hek,hi|²

−^ν₂|h|²

dµ(h)

= R

H

i ∑^ν

j=1

(h,u_j)^∼

exp _ν

j=1∑

(z_j−1) 2z_j |h|²

−^ν₂|h|²

dµ(h)

= R

H

i ∑^ν

j=1

(h,uj)^∼

exp

−¹_{2 ∑}^ν

j=1 1 z_j|h|²

dµ(h)

= Ranw_~z

B^ν δG(~x|~u)dm^ν(~x).

(22)

Corollary 4. For G∈ F(B)and for z∈C⁺and for s.a.e.u∈B, Z _anwz

B δF(_x|u)_dm(_x) = _lim

n→∞zⁿ² Z

Bexp 1−z

2

∑

n j=1

(_ej,x)^∼

2

δG(_x|u)_dm(_x)_{, (23)}

whenever z→ −i q through C⁺.

Now, we prove the change of scale formula for Wiener integrals under the first variation ofG∈ F(B^ν)_.

Theorem 5. For realρ_j>0,j=1, 2,· · ·,ν, R

B^νδG(ρ₁x1,· · ·,ρ_νxν|~u)dm^ν(~x)

= limn→∞

∏^νj=1ρ⁻ⁿ_j

R

B^νexp _ν

j=1∑ ρ²_j−1

2ρ²_j

∑n k=1

(e_k,xj)²

δG(~x|~u)dm^ν(~x), (24) where dm^ν(~x) =dm(x1)dm(x2)· · ·dm(xν).

Proof. In Theorem 4, we have that for realzj>0,j=1, 2,· · ·,ν, R

B^νδG(z₁⁻¹²x₁,· · ·,z_ν⁻¹²x_ν|~u)dm^ν(~x)

= limn→∞

∏^νj=1zj

ⁿ₂ R

B^νexp _ν

j=1∑ (1−z_j)

2

∑n k=1

(ek,xj)²

δG(~x|~u)dm^ν(~x). (25) Takingz_j=ρ⁻²_j ,j=1, 2,· · ·,νin(25), we have(24).

Now, we have a change of scale formula for Wiener integrals under the first variation inF(B^ν).

Corollary 5. For realρ>0and for~u∈B^ν, R

B^νδG(ρx1,· · ·, ,ρxν|~u)dm^ν(~x)

= limn→∞ρ^−νnR

B^νexp _ν

j=1∑

ρ²−1 2ρ²

∑n k=1

(e_k,xj)²

δG(~x|~u)dm^ν(~x). (26)

Proof. Takingzj=s⁻²for allj=1, 2,· · ·,ν in (26), we have the result.

Using (27) forν =1, we have the the change of scale formula for Wiener integrals under the first variation inF(B):

Corollary 6. For G∈ F(B)and for realρ>0and for s.a.e.u∈B, Z

BδG(ρx|u)dm(x) =ρ⁻ⁿ Z

Bexp

ρ²−1 2ρ²

∑

n j=1

(e_j,x)^∼

2

δG(x|u)dm(x). (27)

Finally, we show that the analytic Feynman integral of the first variation inF(B^ν)can be successfully expressed as the limit of a sequence of Wiener integrals of the first variation on(H,B^ν,m).

Theorem 6. Let {zk,m}_m be a sequence of complex numbers from Ω such that {zk,m}_m →

−i qk (_qk 6= 0)_{throughΩ as m} → ∞for k = _{1, 2,}· · ·,ν, where~_q = (_q1,q2,· · ·,qν) ∈ C^ν. Then,

Ran fq

B^ν δG(~x|~u)dm^ν(~x)

= limm→∞

limn→∞

∏^νj=1z_j,m ⁿ₂

R

B^νexp _ν

j=1∑

(1−z_j,m) 2

∑n k=1

(e_k,x_j)^∼ 2

δG(~x|~u)dm^ν(~x)

(28)

where∏^νj=1 zj,m=z1,mz2,m · · ·zν,mand dm^ν(~x) =dm(x1)dm(x2)· · ·dm(xν). Proof. By Definition1,

Ran fq

B^ν δG(~x|~u)dm^ν(~x)

= limm→∞Ranw_~zm

B^ν δG(~x|~u)dm^ν(~x)

= limm→∞R

B^νδG((z1,m)⁻¹²x1,· · ·,(zν,m)⁻¹² x_ν|~u)dm^ν(~x)

= _limm→∞

limn→∞(_∏^ν_j=1z_j,m)ⁿ² R

B^νexp _ν

∑

j=1 (1−z_j,m)

2

∑n k=1

(e_k,x_j)^∼ 2

δG(~x|~u)dm^ν(~x)

(29)

whenever{z_k,m}_m→ −iq_k(q_k 6=0)throughΩasm→_∞fork=1, 2,· · ·,ν.

Corollary 7. Let{zn}be the sequence of complex numbers from C⁺such that{zn} → −iq(q6=0) through C⁺. Then for G∈ F(B)and for s.a.e.u∈B,

Z _{an fq}

B δG(x|u)dm(x) = lim

n→∞zn

n 2

Z

Bexp

1−zn

2

∑

n j=1

(ej,x)^∼

2

δG(x|u)dm(x), (30) whenever{zn} → −iq through C+.

Remark 2. In future works, we will try to prove rather nice formulas than the change of scale formula for the Wiener integral on the Wiener space and the abstract Wiener space and the product abstract Wiener space.

Funding:Fund of this paper was supported by NRF-2017R1A6A3A11030667.

Data Availability Statement:Data sharing not applicable.

Acknowledgments:Fund of this paper was supported by the National Research Foundation Grant, NRF-2017R1A6A3A11030667.

Conflicts of Interest:The author declares no conflict of interest.

References

1. Cameron, R.H.; Martin, W.T. The behavior of measure and measurability under change of scale in Wiener space.Bull. Am. Math.

Soc.1947,53, 130–137. [CrossRef]

2. Cameron, R.H. The translation pathology of Wiener space.Duke Math. J.1954,21, 623–628. [CrossRef]

3. Cameron, R.H.; Storvick, D.A. Relationships between the Wiener integral and the analytic Feynman integral. Supplemento Rendicoti Circolo Matematico Palermo Serie II Numero1988,17, 117–133.

4. Cameron, R.H.; Storvick, D.A. Change of scale formulas for Wiener integral.Supplemento Rendicoti Circolo Matematico Palermo Serie II Numero1987,17, 105–115.

5. Yoo, I.; Skoug, D. A change of scale formula for Wiener integrals on abstract Wiener space. Int. J. Math. Math. Sci.1994,17, 239–248. [CrossRef]

6. Yoo, I.; Yoon, G.J. Change of scale formulas for Yeh-Wiener integrals.Commun. Korean Math. Soc.1991,6, 19–26.

7. Cameron, R.H.; Martin, W.T. On transformations of Wiener integrals under translations.Ann. Math.1944,45, 386–396. [CrossRef]

8. Cameron, R.H.; Martin, W.T. Transformations for Wiener integrals under a general class of linear transformations.Trans. Am.

Math. Soc.1945,58, 184–219. [CrossRef]

9. Cameron, R.H.; Storvick, D.A. Some Banach algebras of analytic Feynman integrable functionals, An analytic functions.Lect.

Notes Math.1980,798, 18–27.

10. Kim, Y.S. A change of scale formula for Wiener integrals about the first variation on the product abstract Wiener space. In Proceeding of the International Conference, “Modern Problems in Mathematics and Mechanics”, Baku, Azerbeijan, 23–25 October 2019; pp. 492–493.

11. Kim, Y.S. Feynman integral and a change of scale formula about the first variation and the Fourier-Stieltjes transform.Mathematics 2020,8, 1666. [CrossRef]

12. Kim, Y.S. Analytic Feynman integral and a change of scale formula for Wiener integral of an unbounded cylinder function.

Complexity2020,2020, 2671474.

13. Kim, Y.S.; Ahn, J.M.; Chang, K.S.; Yoo, I. A change of scale formula for Wiener Integrals on the product abstract Wiener space.J.

Korean Math. Soc.1996,33, 269–282.

14. Kallianpur, G.; Bromley, C. Generalized Feynman Integrals Using Analytic Continuation in Several Complex Variables. In Stochastic Analysis and Applications; Pinsky, M.H., Ed.; Dekker: New York, NY, USA, 1984; pp. 217–267.

15. Gross, L. Abstract Wiener Space. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, 21 June–18 July 1965, 27 December 1965–7 January 1966; pp. 31–42.

16. Kuelbs, J. Abstract Wiener Spaces and Applications to Analysis.Pacific J. Math.1969,31, 433–450. [CrossRef]

17. Kuo, H.H. Gaussian Measures in Banach Spaces. InLecture Notes in Math; Springer: Berlin, Germany, 1975; Volume 463.

18. Chung, D.M. Scale-invariant measurability in abstract Wiener space.Pacific J. Math.1980,130, 27–40. [CrossRef]

19. Cameron, R.H. The first variation of an indefinite Wiener integral.Proc. Am. Soc.1951,2, 914–924. [CrossRef]