Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform

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Article

Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform

Young Sik Kim

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

Received: 20 July 2020; Accepted: 21 September 2020; Published: 28 September 2020 Abstract: We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x) =expRT

0 θ(t,x(t))dt successfully exist under the certain condition, where θ(t,u) = R

Rexp{iuv}dσt(v)is a Fourier–Stieltjes transform of a complex Borel measureσt ∈ M(R)andM(R)is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation ofF(x)sucessfully holds on the Wiener space.

Keywords:Wiener space; Wiener integral; Feynman integral; Fourier–Stieltjes transform; first variation

1. Introduction

1.1. Motivation of the Wiener Integral from the Heat Equation The solution of the heat (or diffusion) equation:

−^∂u

∂t =−¹

24u+V(ξ)u=Hu(ξ∈ R^d, 0≤t),u(0,·) =ψ(·) is of the form:

u(t,ξ) = (e^{−t H}ψ)(ξ) =E

exp

− _Z _t

0 V(x(s) +ξ)ds

ψ x(t) +ξ

, (1)

whereψ∈ L2(R^d),ξ ∈ R^dandx(·)is aR^d−valued continuous function defined on[0,t]such that x(0) = 0; Edenotes the expectation with respect to the Wiener path starting at timet = 0 (Eis the Wiener integral);H = −4+Vis the energy operator (or Hamiltonian),4is a Laplacian, and V:R^d→Ris a potential. (1) is called the Feynman- Kac formula. Applications of the Feynman–Kac formula (in various settings) have been given in the following areas: diffusion equations, the spectral theory of the Schrödinger operator, quantum mechanics, and statistical physics (see [1]).

1.2. Motivation of This Paper

Do the analytic Wiener integral and the analytic Feynman integral about the first variation of the functionF(x) =expRT

0 θ(t,x(t))dt successfully exist?

Is the change of scale formula for the Wiener integral successfully satisfied about the first variation ofF(x) =expRT

0 θ(t,x(t))dt ?

Mathematics2020,8, 1666; doi:10.3390/math8101666 www.mdpi.com/journal/mathematics

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1.3. Research Flow About the Topic of a Change of Scale Formula from 1944

The concept of a transform of Wiener integrals was introduced in [2,3] (1944–1945). The behavior of a measure and a measurability under a change of scale on the Wiener space was expanded in [4] (1947).

The first variation of a Wiener integral was defined in [5] (1951) and and the translation pathology of Wiener integrals was proved in [6] (1954). The functional transform for Feynman integrals similar to Fourier transform was introduced in [7] (1972). Relationships among the Wiener integral and the analytic Feynman Integral was proved in [8] (1987). A change of scale formula for the Wiener integral was proved in [9]. The scale-Invariant Measurability on the Wiener space was proved in [10] (1979).

Theorems about some Banach algebras of analytic Feynman integrable functionals was expanded in [11] (1980).

A change of scale formula for the Wiener integral of the cylinder function on the abstract Wiener space was proved in [12,13]. The relationship between the Wiener integral and the Fourier Feynman transform about the first variation was proved in [14–17]. The behavior of the scale factor for the Wiener integral of the unbounded function on the Wiener space was proved in [18].

1.4. Target of This Paper

The purpose of this paper is the following:

(1) The first variation of a functionF(x) =expRT

0 θ(t,x(t))dt exists under the certain condition.

We find this condition.

(2) The analytic Wiener integral and the analytic Feynman integral of the first variation ofF(x)exist.

(3) A change of scale formula for the Wiener integral holds for the first variation ofF(x)_. 2. Definitions and Preliminaries

LetC0[_0,_T]denote the space of real-valued continuous functionsxon[_0,_T]_{such that}_x(₀) =_0.

LetMdenote the class of all Wiener measurable subsets of C0[0,T]and letmdenote the Wiener measure and(C0[0,T],M,m)be a Wiener measure space, see [1].

The integral of a measurable functionFdefined on the Wiener measure space with respect to a Wiener measurem∈ Mis called a Wiener integral ofFand we denote it byR

C₀[0,T]F(x)dm(x). A subsetEofC0[0,T]is said to be scale-invariant measurable ifρE∈ Mfor eachρ>0, and a scale-invariant measurable set Nis said to be scale-invariant null if m(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.). If two functions FandGare equals−a.e., we writeF ≈ G. For more details, see [1].

Throughout this paper, letRⁿdenote then-dimensional Euclidean space and letC,C+, andC^∼₊ denote the complex numbers, the complex numbers with positive real part, and the non-zero complex numbers with nonnegative real part, respectively.

Definition 1(see [1]). Let F be a complex-valued measurable function on C0[0,T]such that the integral J(F;λ) =

Z

C₀[0,T]F(λ⁻

12x)dm(x) (2)

exists for all realλ >0. If there exists a function J^∗(F;z)analytic onC+such that J^∗(F;λ) = J(F;λ)for all realλ>0, then we define J^∗(F;z)to be the analytic Wiener integral of F over C0[0,T]with parameter z, and for each z∈C+, we write

Z _anw_z

C₀[0,T]F(x)dm(x) =J^∗(F;z). (3)

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Let q be a non-zero real number and let F be a function on C0[0,T]whose analytic Wiener integral exists for each z inC+. If the following limit exists, then we call it the analytic Feynman integral of F over C₀[0,T] with a parameter q and we write

Z _{an f}_q

C₀[0,T]F(x)dm(x) = lim

z→−iq Z _anw_z

C₀[0,T]F(x), (4)

where z approaches−iq throughC+and i²=−1.

Now we introduce the Wiener Integration Formula.

LetC^a,b₀ denote the space of real valued continuous functions x on [a,b] such that x(a) = 0.

Leta=t0<t₁<t2<· · ·<tn≤b.

Theorem 1([1], Theorem 3.3.5). Let f :Rⁿ →R be Lebesgue measurable. Then we have Z

C^a,b₀ f(x(t1),x(t2),· · ·,x(tn))dm(x)

= _n

∏

j=1

2π(t_j−tj−1) −¹₂ Z

Rⁿ f(~u)exp

−¹ 2

∑

n j=1

(u_j−uj−1)² tj−tj−1

d~u, (5)

where u₀=0and where the equality is in the strong sense that if either side of (5)is defined(in the sense of the Lebesgue theory), whether finite or infinite, then so is the other side and they agree.

Remark 1. If we let a=0and b=T in(5), we have the following Wiener integration formula. Let C0[0,T]be a Wiener space and let0≤t₁≤t2≤ · · · ≤tn≤T. Then

Z

C₀[0,T] f(x(t₁),x(t₂),· · ·,x(tn))dm(x)

= n

∏

j=1

2π(tj−tj−1) −¹₂ Z

Rⁿ f(~u)exp

−¹ 2

∑

n j=1

(uj−uj−1)² t_j−tj−1

d~u, (6)

where f : Rⁿ → R is a Lebesgue measurable function and ~u = (u1,u2,· · ·,un) and u0 = 0 and d~u=du1du2· · ·dun.

Definition 2([5]). Let F be a measurable function on C0[0,T]. Then for w∈C0[0,T], δF(x|w) = ^∂

∂hF(x+hw)|_h=0 (7)

is called the first variation of F in the direction w∈C0[0,T](if it exists).

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

Z

Rexp{−au²+ibu}du= rπ

a exp{−^b

2

4a}, (8)

whereais a complex number withRe a>0 andbis a real number andi²=−1.

Theorem 2([19], Morera’s Theorem). Suppose f is a complex function in an open setΩsuch that Z

∂∆ f(z)dz=0, (9)

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for every closed triangle∆⊂_{Ω. Then f} ∈H(_Ω), where H(_Ω)is the class of all analytic functions in an open setΩand∂∆is a boundary of∆.

3. The Main Results

Define a function

F(x) =exp{ Z T

0 θ(t,x(t))dt}, (10)

which has been used in the Quantum Mechanics, (see [1]).

Definition 3([1,18]). Letθ:[_0,T]×R→Cbe defined by θ(t,u) =

Z

Rexp{iuv}dσt(v), (11)

which is a Fourier–Stieltjes transform of a complex Borel measureσt∈M(R), whereM(R)is a set of complex Borel measures defined on R.

Notation 1.

(1) Let∆nbe defined by

∆n(T)≡ {(t₁,t₂,· · ·tn)|0≤t₁≤t₂≤ · · · ≤tn ≤T},t₀=₀(₁≤n)_.

(2) Let M_n⁰⁰ = M⁰⁰_n(_∆_n(T)×Rⁿ)be the set of a countably additive complex Borel measure defined on the product space∆n(T)×Rⁿ(1≤n), see [1].

Throughout this paper, we suppose that~_v = (_v₁_,_v₂_,· · ·,vn)_and_w ∈ C0[_0,_T] _with|w| < _∞ and∑^∞n=1||µ_n||<∞, where|| · ||is a total variation of a countably additive complex Borel measure, and we suppose that∑ⁿj=1|v_j|<_∞and

∑

∞ n=1

Z

∆n(T)×Rⁿ

_n

j=1

∑

|v_j|

d|µn|(~_t,~v)<_∞. (12)

Lemma 1([20]). Let F:C0[0,T]→Cbe defined by(10)and(11). Then we have that F(x) =

∑

∞ n=1

Z

∆n(T)×Rⁿ exp

i

∑

n j=1

vjx(tj)

dµn(~_t,~v), (13) where µ_n is a countably additive Borel measure defined on ∆n(T)× Rⁿ for each n = 1, 2,· · ·, and dµn(~_t,~v) =

∏ⁿj=1dσt_j(vj)dtj

.(For more details, see [20]).

First we calculate the first variation ofF(x)and we find a condition of the existence of the first variation ofF(x).

Lemma 2. The first variation of F(x)in(10)exists under the condition(12)and is of the form : δF(x|w) =

∑

∞ n=1

Z

∆n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

exp

i

∑

n j=1

v_jx(t_j)

dµn(~_t,~v). (14)

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Proof. By the definition of the first variation in [5] and Dominated Convergence Theorem, we have that δF(x|w)≡ ^∂

∂hF(x+hw)|_h=0

= lim

h→0

4h→0lim 1 4h

F(x+ (h+4h)w)−F(x+hw)

= _lim

h→0

4h→0lim 1 4h

_∞

n=1

∑

Z

∆n(T)×Rⁿexp

i

∑

n j=1

v_j(x(t_j) + (h+4h)w(t_j) )

−

∑

∞ n=1

Z

∆n(T)×Rⁿexp

i

∑

n j=1

v_j(x(t_j) +h w(t_j) )

dµn(~_t,~v)

= _lim

h→0

4h→0lim 1 4h

_∞

n=1

∑

Z

∆n(T)×Rⁿexp

i

∑

n j=1

v_j(x(t_j) + h w(t_j) )

×

exp

i4h

∑

n j=1

v_jw(t_j)

−1

dµn(~_t,~v)

= _lim

h→0

_∞

n=1

∑

Z

∆n(T)×Rⁿ exp

i

∑

n j=1

v_j(x(t_j) + h w(t_j) )

×

4h→0lim exp

i4h ∑ⁿ_j=1vjw(tj)

−1

4h dµn(~_t,~_v)

= lim

h→0

_∞

n=1

∑

Z

∆n(T)×Rⁿ exp

i

∑

n j=1

vj(x(tj) + h w(tj) )

×

4h→0lim

i ∑ⁿj=1vjw(tj)

exp

i4h∑ⁿj=1 vjw(tj)

1 dµn(~_t,~v)

=

∑

∞ n=1

Z

∆n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp

i

∑

n j=1

vjx(tj)

dµn(~_t,~v), (15)

where we use L’Hospital’s Rule in the fifth equalty. Forw∈C0[0,T]with|w|<∞,

|δF(x|w)| ≤

∑

∞ n=1

Z

∆n(T)×Rⁿ

∑

n j=1

v_jw(t_j)

d µn

(~_t,~v)

≤

∑

∞ n=1

Z

∆n(T)×Rⁿ |w|

∑

n j=1

|v_j| d µn

(~_t,~v)

= |w|

∑

∞ n=1

Z

∆n(T)×Rⁿ

n

∑

j=1

|v_j|

d µn

(~_t,~v)_. ₍₁₆₎ Therefore, the first variationδF(x|w)ofF(x)exists under the condition (12).

Now, we prove the existence of the analytic Wiener integral of the first variation ofF(x)in (10).

Theorem 3. For z∈C⁺, the analytic Wiener integral of the first variation of F(x)in(10)exists and is given by Z anw_z

C₀[0,T] δF(x|w)dm(x)

=

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp

− ¹ 2z

∑

n j=1

(tj−tj−1)

∑

n k=j

v²_k

dµn(~_t,~v). (17)

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Proof. By Fubini’s Theorem, Wiener integration formula (6) and Lemma2, we have that for realλ>0 and forw∈C₀[0,T],

g(λ) = Z

C₀[0,T] δF(λ⁻

12x|w)dm(x)

= Z

C₀[0,T]

_∞

n=1

∑

Z

4_n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

exp

iλ⁻¹²

∑

n j=1

v_jx(t_j)

dµn(~_t,~v)

dm(x)

=

∏

n j=1

( ^λ

2π(t_j−tj−1))¹² Z

Rⁿ

_∞

n=1

∑

Z

4(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

×exp

− ^λ 2

∑

n j=1

[u_j−uj−1]² t_j−tj−1

+i

∑

n j=1

u_jv_j

dµn(~_t,~v)

d~u

=

∑

∞ n=1

Z

4(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

× _Z

Rⁿ

n

∏

j=1

( ^λ

2π(t_j−tj−1))¹²

exp

−^λ 2

∑

n j=1

+i

∑

n j=1

ujvj

d~u

dµn(~_t,~v)

=

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp

− ¹ 2λ

∑

n j=1

(tj−tj−1)

∑

n k=j

v²_k

dµn(~_t,~v), (18) where forλ∈C⁺,

Z

C₀[0,T] exp

iλ⁻

1 2

∑

n j=1

vjx(tj)

dm(x)

=

∏

n j=1

( ^λ

2π(tj−tj−1))¹² Z

Rⁿ exp

− ^λ 2

∑

n j=1

[uj−uj−1]² tj−tj−1

+ i

∑

n j=1

ujvj

d~u

= exp

− ¹ 2λ

∑

n j=1

(tj−tj−1)

∑

n k=j

v²_k

. (19)

We use the following function in the second equality of (18), when we applied the Wiener integration formula (6):

f(x(t1),x(t2),· · ·,x(tn)) =

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp

iλ⁻¹²

∑

n j=1

vjx(tj)

dµn(~_t,~v)

.

Now we prove the existence of the analytic Wiener integral of the first variationF(x)in (10).

Forz∈C⁺, let g^∗(z) =

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

exp

− ¹ 2z

∑

n j=1

(t_j−tj−1)

∑

n k=j

v²_k

dµn(~_t,~v). (20)

Theng^∗(λ) =g(λ)for all realλ>0. Forz∈C⁺,

|g^∗(z)| ≤

∑

∞ n=1

Z

4_n(T)×Rⁿ

|w|

∑

n j=1

|v_j| exp

− ^z^¯ 2z¯z

∑

n j=1

(t_j−tj−1)

∑

n k=j

v²_k

d|µn|(~_t,~v)

≤

∑

^∞

n=1 Z

4_n(T)×Rⁿ

|w|

∑

n j=1

|vj|

d|µn|(~_t,~v)

= |w|

∑

^∞

n=1 Z

4_n(T)×Rⁿ

n j=1

∑

|vj|

d|µn|(~_t,~v)<∞, (21)

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becauseRe(z) =Re(z¯)>0. By Dominated Convergence Theorem,g^∗(z)is a continuous function of z∈C⁺.

Since the function : exp

−_2z¹ _∑ⁿ_j=1(_t_j−tj−1)_∑ⁿ_k=j_v²_k is an analytic function ofz∈C⁺for each

~v= (v1,v2,· · ·,vn)∈Rⁿ, we have thatR

Γexp

−_2z¹ _∑ⁿ_j=1(tj−tj−1) _∑ⁿ_k=jv²_k dz=0 for all rectifiable simple closed curvesΓinC⁺by Cauchy Integral Theorem.

As for allz∈C⁺,

i

∑

n j=1

v_jw(t_j)

exp

− ¹ 2z

∑

n j=1

(t_j−tj−1)

∑

n k=j

v²_k

≤ |w|

∑

n j=1

|v_j|<_∞, (22)

we can apply Fubini’s Theorem to the integralR

Γg^∗(z)dzand we haveR

Γg^∗(z)dz=0.

By Morera’s Theorem in [19],g^∗(z)is an analytic function ofzinC⁺. Therefore the analytic Wiener integral of the first variation ofF(_x)successfully exists and is given by (17) forz∈C⁺and for w∈C0[0,T].

Remark 2. In(19), we use the formula : For realλ>0and for each j=1, 2,· · ·,n,

( ^λ

2π(tj−tj−1))¹² Z

Rexp

−^λ[uj−uj−1]²

2(tj−tj−1) +iujw)

duj

= exp

iuj−1w− (t_j−tj−1)w² 2λ

, (23)

which is proved by the transformation z= [_(t ^λ

j−t_j₋₁]¹² (uj−uj−1)and by(8).

Now, we obtain the analytic Feynman integral of the first variationδF(x|w)of functions F : C0[0,T]→Cin (10) and prove the existence of the analytic Feynman integral.

Theorem 4. The analytic Feynman integral of the first variation of F(x)in(10)exists and Z _{an f}_q

C₀[0,T]δF(x|w)dm(x)

=

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

exp

− ⁱ 2q

∑

n j=1

(t_j−tj−1)

∑

n k=j

v²_k

dµn(~_t,~v). (24)

Proof. Form=1, 2, 3· · · and forzm∈C⁺, let fm(~v) =

i

∑

n j=1

vjw(tj)

exp

− ¹ 2zm

∑

n j=1

(tj−tj−1)

∑

n k=j

u²_k

(25)

and letzm→ −iqwheneverm→_{∞. Then} fm(~v)→ f(~v), where f(~v) =

i

∑

n j=1

vjw(tj)

exp

− ⁱ 2q

∑

n j=1

(tj−tj−1)

∑

n k=j

u²_k

(26)

and|fm(~v)| ≤ |w| _∑ⁿ_j=1|vj|<∞for allm=1, 2,· · ·.

(8)

By Dominated Convergence Theorem and Theorem3, the analytic Feynman integral of the first variation ofF(x)in (10) exists and

Z an f_q

= lim

zm→−iq Z _anw_zm

= lim

z_m→−iq

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

exp{− ¹ 2zm

∑

n j=1

(t_j−tj−1)

∑

n k=j

u²_k

dµn(~_t,~v)

= lim

m→∞

∑

∞ n=1

Z

4_n(T)×Rⁿ fm(~v)dµn(~_t,~v)

=

∑

∞ n=1

Z

4_n(T)×Rⁿ f(~v)dµn(~_t,~v)

=

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp{− ⁱ 2q

∑

n j=1

(tj−tj−1)

∑

n k=j

u²_k

dµn(~_t,~v), (27)

To prove harmonious relationships among the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation and to prove the change of scale formula for the Wiener integral of the first variation ofF(x)in (10), we first have to prove the following property:

Theorem 5. For z∈C⁺, the function exp

1−z 2

∑

n j=1

[x(tj)−x(tj−1) ]² tj−tj−1

δF(x|w) (28)

is a Wiener integrable function of x∈C0[0,T].

(9)

Proof. By Fubini Theorem, Wiener integration formula (6), (12), (14) and (19), we have that forz∈C⁺,

Z

C₀[0,T]exp 1−z

2

∑

n j=1

δF(x|w)dm(x)

= Z

C₀[0,T]exp 1−z

2

∑

n j=1

× _∞

n=1

∑

Z

4_n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp

i

∑

n j=1

vjx(tj)

dµn(~_t,~v)

dm(x)

=

∏

n j=1

[2π(tj−tj−1)]⁻¹² Z

Rⁿexp 1−z

2

∑

n j=1

[uj−uj−1)]² tj−tj−1

× _∞

n=1

∑

Z

4_n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp

i

∑

n j=1

ujvj

dµn(~_t,~v)

×exp

−¹ 2

∑

n j=1

[uj−uj−1)]² tj−tj−1

d~_u

=

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

∏

n j=1

[2π(t_j−tj−1) −¹₂

× Z

Rⁿexp

− ^z 2

∑

n j=1

[u_j −uj−1) ]² t_j−tj−1

+i

∑

n j=1

u_jv_j

d~u

dµn(~_t,~v)

=

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

z⁻ⁿ²

× n

∏

j=1

z 2π(t_j−tj−1)

ⁿ₂ Z

Rⁿexp

− ^z 2

∑

n j=1

[uj−uj−1)]² t_j−tj−1

+i

∑

n j=1

ujvj

d~u

dµn(~_t,~v)

= z⁻ⁿ²

∑

n j=1 Z

4_n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp

− ¹ 2z

∑

n j=1

(tj−tj−1)

∑

n k=j

v²_k

dµn(~_t,~v)

= z⁻ⁿ²

∑

n j=1 Z

4_n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

exp

− ^z^¯ 2zz¯

∑

n j=1

(t_j−tj−1)

∑

n k=j

v²_k

dµn(~_t,~v)

≤ |z|⁻ⁿ² _∞

n=1

∑

Z

4_n(T)×Rⁿ(|w|

∑

n j=1

|v_j|)×1d|µn|(~_t,~v)

= |z|⁻ⁿ² |w| _∞

n=1

∑

Z

4_n(T)×Rⁿ

∑

n j=1

|v_j|d|µn|(~_t,~v)

< ∞, (29)

because forz∈C⁺,

exp

− ^z^¯ 2zz¯

∑

n j=1

(t_j−tj−1)

∑

n k=j

v²_k

≤1.

(10)

We use the following function in the second equality of (29), when we applied the Wiener integration formula (6):

f(x(t₁)_,x(t₂)_,· · ·.x(tn))

=exp 1−z

2

∑

n j=1

× _∞

n=1

∑

Z

4_n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp

i

∑

n j=1

vjx(tj)

dµn(~_t,~v). (30) Therefore we have the desired result.

Remark 3. (1) In [14], Y.S. Kim proved the relationship between the Fourier-Feynman transform and the Wiener integral about the first variation of

F(x) = Z

Hexp

i(h,x)^∼

dµ(h) in the Fresnel class under the condition that

Z

H

|h|d|µ|(h)<_∞ on the abstract Wiener space.

(2) In [17], Y.S. Kim proved the relationship between the Fourier-Feynman transform and the convolution about the first variation of

F(x) =µˆ((h1,x)^∼,· · ·,(hn,x)^∼) = Z

Rⁿexp

i

∑

n j=1

(hj,x)^∼vj

µ(d~v) under the condition that

Z

Rⁿ

n j=1

∑

v²_j ¹₂

|µ|(d~v)<_∞ on the abstract Wiener space.

(3) In this paper, we find a condition :

∑

∞ n=1

Z

∆n(T)×Rⁿ

_n

j=1

∑

|v_j|

d|µn|(~_t,~v)<_∞

that the first variation of F(x)in(10)exists, where F(x) =exp{

Z _T

0 θ(t,x(t))dt}=

∑

∞ n=1

Z

∆n(T)×Rⁿ exp

i

∑

n j=1

vjx(tj)

dµn(~_t,~v) on the Wiener space.

Now we establish the harmonious relationship between the analytic Wiener integral and the Wiener integral of the first variation.

That is, we prove that the analytic Wiener integral of the first variation can be successfully expressed by the sequence of Wiener integrals of the first variation.

(11)

Theorem 6. For z∈C⁺, Z _anw_z

C₀[0,T] δF(x|w) dm(x)

= zⁿ² Z

C₀[0,T]exp 1−z

2

∑

n j=1

[x(t_j)−x(tj−1) ]² t_j−tj−1

δF(x|w)dm(x). (31) Proof. By Wiener integration formula (6), Lemma2, Theorem3and Theorem5, we have that for z∈C⁺and forw∈C0[_0,_T]_,

Z

C₀[0,T]exp 1−z

2

∑

n j=1

[x(t_j)−x(tj−1) ]² t_j−tj−1

δF(x|w)dm(x)

= Z

C₀[0,T]exp 1−z

2

∑

n j=1

[x(tj)−x(tj−1) ]² t_j−tj−1

× _∞

n=1

∑

Z

∆n(T)×Rⁿ

i

∑

n j=1

vjw(tj)

exp

i

∑

n j=1

vjx(tj)

dµn(~_t,~v)

dm(x)

=

∏

n j=1

2π(t_j−tj−1) ⁻¹₂ Z

Rⁿ exp 1−z

2

∑

n j=1

× _∞

n=1

∑

Z

∆n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

exp

i

∑

n j=1

v_ju_j)

×exp

−¹ 2

∑

n j=1

dµn(~_t,~v)

d~u

= z⁻ⁿ² _n

∏

j=1

z 2π(t_j−tj−1)

¹₂ _Z

Rⁿ exp

−^z 2

∑

n j=1

× _∞

n=1

∑

Z

∆n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

exp

i

∑

n j=1

v_ju_j

dµn(~_t,~v)

d~u

= z⁻ⁿ²

∑

∞ n=1

Z

4_n(T)×Rⁿ

i

∑

n j=1

v_jw(t_j)

exp

− ¹ 2z

∑

n j=1

(t_j−tj−1)

∑

n k=j

u²_k

dµn(~_t,~v)

= z⁻ⁿ² Z _anw_z

C₀[0,T]δF(x|w)dm(x). (32)

Now, we prove that the first variation ofF(x)satisfies successfully the change of scale formula for the Wiener integral on the Wiener space.

Theorem 7. For positive realρ > 0, Z

C₀[0,T]δF(ρx|w)dm(x)

= ρ⁻ⁿ Z

C₀[0,T]exp

ρ²−1 2ρ²

∑

n j=1

[x(tj)−x(tj−1) ]² t_j−tj−1

δF(x|w)dm(x). (33)

(12)

Proof. By Theorem6, we have that for realz>0, Z

C₀[0,T]δF(z⁻¹²x|w)dm(x)

= zⁿ² Z

C₀[0,T]exp 1−z

2

∑

n j=1

[x(t_j)−x(tj−1)]² t_j−tj−1

δF(x|w)dm(x). (34)

If we letz= ρ⁻²in the above equation, we can have the desired result.

Now, we establish the harmonious relationship between the analytic Feynman integral and the Wiener integral of the first variation ofF(x).

That is, we prove that the analytic Feynman integral of the first variation can be successfully expressed as the limit of the sequence of Wiener integrals of the first variation on the Wiener space.

Theorem 8.

Z _{an f}_q

C₀[0,T]δF(x|w)dm(x)

= lim

k→∞zk

n 2

Z

C₀[0,T]exp

1−zk

2

∑

n j=1

[x(tj)−x(tj−1)]² tj−tj−1

δF(x|w)dm(x), (35)

whenever{zk} → −i q throughC+.

Proof. By Definition1, Lemma2and Theorem6, we have that Z an f_q

C₀[0,T] δF(x|w) dm(x)

= _lim

k→∞ Z _anw_zk

C₀[0,T] F(x)dm(x)

= lim

k→∞z_kⁿ² Z

C₀[0,T]exp

1−z_k 2

∑

n j=1

[x(t_j)−x(tj−1)]² tj−tj−1

δF(x|w)dm(x), (36) whenever{z_k} → −i qthroughC+.

4. Conclusions and Discussion

The first variation ofF(x) =_expRT

0 θ(t,x(t))dt successfully exists under the condition (12).

The Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation ofF(x)successfully exist. Furthermore, the first variation ofF(x)successfully satisfies the change of scale formula for the Wiener integral.

The Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x) exist under the condition that ∑^∞n=1||µ_n|| < ∞ and

∑^∞n=1

R

∆n(T)×Rⁿ

∑ⁿj=1|v_j|

d|µn|(~_t,~v)<_∞.

(13)

But the Wiener integral, the analytic Wiener integral and the analytic Feynman integral ofF(x) exist and all properties ofF(x)hold under one condition that∑^∞n=1||µn||<_∞:

(1). Z

C₀[0,T]F(x)dm(x) =

∑

∞ n=1

Z

4_n(T)×Rⁿexp

−¹ 2

∑

n j=1

(t_j−tj−1)

∑

n k=j

u²_k

dµn(~_t,~v) (2).

Z _anw_z

C₀[0,T]F(x)dm(x) =

∑

∞ n=1

Z

4_n(T)×Rⁿexp

− ¹ 2z

∑

n j=1

(t_j−tj−1)

∑

n k=j

u²_k

dµn(~_t,~v) (₃)_.

Z _{an f}_q

C₀[0,T]F(x)dm(x) =

∑

∞ n=1

Z

4_n(T)×Rⁿexp

− ⁱ 2q

∑

n j=1

(t_j−tj−1)

∑

n k=j

u²_k

dµn(~_t,~v)

(4). Z _anw_z

C₀[0,T]F(x)dm(x) =zⁿ² Z

C₀[0,T]exp 1−z

2

∑

n j=1

F(x)dm(x) (5).

Z

C₀[0,T]F(ρx)dm(x) =ρ⁻ⁿ Z

C₀[0,T]exp

ρ²−1 2ρ²

∑

n j=1

F(x)dm(x) (6).

Z an f_q

C₀[0,T]F(x)dm(x)

= lim

k→∞z_kⁿ² Z

C₀[0,T]exp

1−zk

2

∑

n j=1

[x(tj)−x(tj−1)]² t_j−tj−1

F(x)dm(x), (37)

because|R

C₀[0,T]F(x)dm(x)| ≤_∑^∞_n=1R

4_n(T)×Rⁿ 1d|µn|(~_t,~v) =_∑^∞_n=1||µn||<∞. That is, (₁)_.

Z

C₀[0,T]F(x)dm(x)

≤

∑

∞ n=1

||µn||<_∞ (2).

Z anw_z

C₀[0,T]F(x)dm(x)

≤

∑

^∞

n=1

||µn||<_∞ (3).

Z _{an f}_q

C₀[0,T]F(x)dm(x)

≤

∑

^∞

n=1

||µ_n||<∞. (38)

Example 1. Let F:C₀[_0,T] →Cbe defined by F(x) =_exp

i∑ⁿ_j=1v_jx(t_j)

with∑ⁿ_j=1|v_j|< _∞and let 4n(T)be defined as in Notation1. Then for w∈C0[0,T]with|w|<∞,

δF(x|w) =

i

∑

n j=1

v_jw(t_j)

exp

i

∑

n j=1

v_jx(t_j)

and|δF(x|w)| ≤ |w|

∑ⁿj=1|vj|

< _∞,using the proof of Lemma2. The analytic Wiener integral and the analytic Feynman integral exist and for z∈C⁺,

Z _anw_z

C₀[0,T]δF(x/w)dm(x) =

i

∑

n j=1

v_jw(t_j)

exp

− ¹ 2z

∑

n j=1

(t_j−tj−1)

∑

n k=j

v²_k

and

Z _{an f}_q

C₀[0,T]δF(x/w)dm(x) =

i

∑

n j=1

v_jw(t_j)

exp

− ⁱ 2q

∑

n j=1

(t_j−tj−1)

∑

n k=j

v²_k

whenever z→ −iq throughC⁺,using the proof of Theorems3and4. Furthermore, we can prove Theorems5–8 about the first variation of F(x)exploiting proofs of those Theorems.

(14)

Funding:This paper is supported by NRF-2017R1A6A3A11030667 in the project from 2017. NRF supported me the research fund for papers listed in [12–18,20] since 1996.

Acknowledgments:I am very gratitude to professors, R. H. Cameron and D. A. Storvick and W. T. Martin and G.

W. Johnson and D. L. Skoug for me to research about the structure of the change of scale formula and about very harmonious relationships among the Wiener integrala and the analytic Wiener integral and the analytic Feynman integral in [2–6,8–11]. Furthermore, I am very gratitude to professors, G. W. Johnson and M. L. Lapidus for me to research about the Feynman integral theory from their book in [1].

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

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c

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