Behavior of a First Variation under an Analytic
Feynman Integral and a Convolution
Young Sik Kim
Department of Mathematics, Research Institute of Natural Sciences, Industry -University Cooperation Foundation,
Hanyang University,
Wangsimni-ro, Seongdong-gu, Seoul 04763, Korea.
Abstract—We investigate the behavior of a first variation under an analytic Feynman integral and a convolution for cylinder functions , ~, ⋯ , , ~ ) and we prove that the
analytic Wiener integral and the analytic Feynman integral of the convolution of two cylinder functions can be successfully expressed as the product of two analytic Wiener integrals and two analytic Feynman integrals.
Keywords—wiener space; wiener integral; feynman integral; fourier-stieltzes transform
I. INTRODUCTION
In [3] and [4], various formulas for linear transformations of Wiener integrals have been given, but Wiener measure was known to behave so badly under the change of scale integral transformations in [1].
In [7], R.H. Cameron and D.A. Storvick established a relationship between the Wiener integral and the analytic Feynman integral, where they expressed the analytic Wiener and Feynman integrals as limits of Wiener integrals for certain Banach algebra of functionals which was given in [5]. Using these results, theyfound a change of scale integral transformation formula for Wiener integrals on a classical Wiener space C 0, in [6]. In [11], G.W.Johnson and D. Skoug established the scale invariant measurability in Wiener space and in [2] D.M.Chung prove thescale-invariant measurability in abstract Wiener spaces.
In [13], Kim proved the change of scale formula about the cylinder function , ~, ⋯ , , ~ ), where
∈ , 1 ∞ and in [14]~[15], Kim proved relationships among Wiener integrals and Feynman integrals and Fourier Feynman transforms and convolutions and first variations for , ~, ⋯ , , ~ ), where is a Fourier transform of a measure μ ∈ .
In this paper, we prove some relationships among the Wiener integral and the Feynman integral and the convolution about the first variation of , ~, ⋯ , , ~ ), where ∈ , 1 ∞ and we prove that the analytic Wiener integral and the analytic Feynman integral of the convolution of two cylinder functions can be successfully expressed as the product of two analytic Wiener integras and two analytic Feynman integrals.
II. DEFINITIONS AND PRELIMINARIES Let H be a real separable infinite dimensional Hilbert space with inner product ∙,∙ and norm | ∙| √ ∙,∙ . Let || ∙ || be a measurable norm on H with respect to the Gauss measure μ ∈ . Let B denote the completion of H with respect to || ∙ ||. Let i denote the natural injection from H into B. The adjoint operator ∗ of i is one-to-one and maps ∗ continuously onto a dense subset of ∗, where ∗ and ∗ are topological duals of H and B, respectively. By identifying H with ∗ and ∗ with ∗ ∗ , we have a triplet ∗, H, B such that ∗ ∁ ∗≡ ∁ and h, x h, x for all in ∗ and in H, where ∙,∙ denotes the natural dual pairing between ∗ and B. By a well known result of Gross [1], ∙ has a unique countably additive extension m to the Borel -algebra B) on B. Then (B,H,m) is
called an abstract Wiener space and m is called a Wiener measure. We denote the Wiener integral of a functional F by
. (1)
Let denote a complete orthonormal system in H such that s are in ∗. For each ∈ and x ∈ B , we define a stochastic inner product ∙,∙ ~ between H and B as follows :
, ~ lim→ ∑ 〈 , 〉 , ,
0 , . (2) It is well known that for every h∈ , , ~exists for μ a. e. x in B and it has a Gaussian distribution with mean zero and variance | | . Furthermore, it is easy to show that , ~ , for μ a. e. x in B if h ∈ ∗, , ~ is essentially independent of the complete orthonormal set used in its definition, and finally that if , ~, ⋯ , , ~ is an orthonormal set of elements in H, then
with mean zero and variance one. Note that if both h and x are in H, then , ~ 〈 , 〉 .
Throughout this paper, let denote the n -dimensional Euclidean space and let C, , ~ denote the complex numbers, the complex numbers with positive real part, and the non-zero complex numbers with nonnegative real part, respectively.
Definition 2.1.Let (B,H,m) be an abstract Wiener space.
Let | 0 and ~ | 0 .
Let F be a complex-valued scale invariant measurable function on B such that the integral
(3)
exists for all real r 0. If there exists an analytic function
∗ analytic on such that ∗ for all real
0, then we define ∗ to be the analytic Wiener integral of F over B with parameter ∈ and for each ∈ , we write
∗ (4)
Let q be a non-zero real number and let F be a function whose analytic Wiener integral exists for each ∈ . If the limit exists, then we call it the analytic Feynman integral of F over B with parameter q, and we write
lim
→ (5)where z approaches iq through and 1. The following is a Wiener integration formula for the Wiener integral on the abstract Wiener space
.
Theorem 2.3. Let Let (B,H,m) be an abstract Wiener
space. and let F be a function on B of the
form , ~, ⋯ , , ~ ), where ∶ → is a
Lebesgue measurable function. Then
, ~, ⋯ , , ~
∑ d (6)
where " " means that if either side exists, then both sides exists and they are equal.
Now, we define the first variation of F in the direction w∈ B on the abstract Wiener space.
Theorem 2.4. Let F be functions on the abstract Wiener space. Then for ∈ , the function of the form :
δ | Ə
Ə |Һ (7)
is called the first variation of F in the direction ∈ (if it exists).
Remark. In the next section, we will several times the following formula :
| | (8)
where a is a complex number with Re(a) > 0, and b is a real number and 1.
III. THE MAIN RESULT
In this section, we establish some relationships among analytic Feynman integral, analytic Wiener integrals and Wiener integrals about the first variation for functions of the form:
, ~, ⋯ , , ~ ) (9)
where ∶ → is in ) , 1
∞
and , ~, ⋯ , , ~ is an orthonormal set of elements in H.Let F(n:p) be the class of functions of the form (9), where f( )∈ ,
1
∞
and F(n,
∞
) be the class of cylinder functions, where f( ) is continuous and bounded.Let C(n:p) be the class of functions of the form (9), where Ə ,⋯ , ,
Ə , 1
and ∈ ,
1
∞
1 . Let C(n,
∞
) be the class of cylinder functions, where, 1
is continuous and bounded.
First, we introduce the first variation and the analytic Wiener integral and the analytic Feynman integral about the first variation of F(x) in (9) on the abstract Wiener space.
Lemma 3.1. Let F be a function of the form (9), where F ∈ C(n:p), 1
∞
and w∈ H. Then for ∈ , the first variation of F is| ∑ , ~ · , ~, ⋯ , , ~ (10)
Proof. See the proof of Theorem 3.1 in [14].
Theorem 3.2. Let F be a function of the form (9), where F ∈ C(n:p), 1 ∞ and w∈ H.
I. When 1 ∞: For every z ∈ and for ∈ , the analytic Wiener integral of the first variation is
|
∑ , ~ ∑ d (11)
where ∈ , 1 ∞ 1 .
II. When p=1. For w∈ H and for every ∈ , the analytic Feynman integral of the first variation is
| ∑ , ~ ∑ d (12) Proof. For all real 0 and for w∈ H, | , ~ , ~ , ⋯ , , ~ ∑ , ~ ∑ d . (13)
Let δ | , ~, ⋯ , , ~ . For all real ∈ R
and for w∈ B, let
| : ) = , ~ , ⋯ , , ~ | dm(x) = dm(x) . (14)
For w∈ H and for every ∈ ,
∗ | :
∑ , ~ ∑ dz. (15)
Then for w∈ H and for all real 0,
∗ | : | : . (16) We will use Morera's Theorem to show that (16) is an analytic function of ∈ w∈ H. First of all, by the Dominated Convergence Theorem, we can show that w∈ H, (16) is a continuous function of .
Let C be any rectifiable simple closed curve lying in . We need only to show that for w∈ H, ＊ (δF(·|w ) : z )dz
= 0
. We shall use the Cauchy Integral Theorem to deduce this
result. Let sup : ∈ C inf : ∈
C .
Case I. If F∈ C(n:1), then for w∈ H, the function
∑ | | · | w | · | | dominates the function
| | | ∑
, ~ | ∑ and
is an integrable function of ∈ , because ∈ .
Case II. If F∈ C(n:p), for 1 ∞, then for w∈ H,
the function ∑ | | · | w | · | | ∑ dominates thr function | | 2 | , ~ | 2
and is an integrable function of ∈ by Holder inequality, because ∈ and for 1 1` 1 ,
∑ ∈ .
Case III. If F∈ C(n: ∞), then is bounded for
1 .and the function
∑ | | · | w | · | | ∑ dominates | | 2 | , ~ | 2
and is an integrable function of ∈ ,
| ∑ ∈ 1
Hence, we can apply the Fubini Theorem to the integral
＊ (δF(·|w ) : z )dz and we have that
＊ (δF(·|w ) : z )dz = 0 for w∈ H , because the function
2 ∑ 2
1
is an analytic functionof z in .for w∈ H. Then we have the desired result.
Now, we establish some relationships among the analytic Feynman integral and the convolution for functions :
, ~, ⋯ , , ~ )
where ∶ → and ∶ → are in ) and 1, ~, ⋯ , , ~ is an orthonormal set of elements in
H, | ＊ | ≡
1, ~, ⋯ , , ~ ,.
First we calculate the convolution of two first variation of two cylinder functions F(x) and G(x) :
Theorem 3.3. For ∈ , the convolution of two first
variation of F(x) and G(x) is given
| ＊ | ≡ , ~, ⋯ , , ~ (18) ,where for ∈ and for , , ⋯ , 2 , ~ √2 ∑ , ~ √ · ∑ d . (19)
Moreover, for ∈ and for ∈ H, | ＊
| ∈ : 1 and | | | |2 | | | | || | | · |∑ | | || | | (20) Proof. By the definition of the convolution of F and G, we have that for real 0 and for s‐a.e. ∈B, | ＊ | √ √ dm(x) = ∑ , ~ 1, √ ~ , ⋯ , , √ ~ , ~ , √2 ~ , ⋯ , , √2 ~ = ∑ , ~ √ 1 , ~ 1 ,⋯ ,√ 1 , ~ ] · ∑ , ~ √ 1 , ~ 1 ,⋯ ,√ 1 , ~ ∑ d . (21)
By analytic continuation in ∈ , we have that for ∈ and s‐a.e. ∈B, | ＊ | √ | √ | dm(x) = ∑ , ~ 1 √ , ~ 1 ,⋯ ,√ 1 , ~ ] · ∑ , ~ √ 1 , ~ 1 ,⋯ ,√ 1 , ~ ∑ d . = , ~, ⋯ , , ~ , (22)
where for ∈ and for , , ⋯ , ,
2 , ~
√2
∑ , ~ √ ∑ d . (23)
Using the following transformation :
√ , √ , we have that for ∈ and for and s‐a.e. ∈B, d 2 〖 , ~ √2 | | , ~ √2 | d d 〖 ∑ , ~ | ∑ , ~ d d 2 ∙ | | ∙ d 1 ∙| |∙ d ∑ ∙ | | ∙ 1 ∑ 1 ∙| |∙ . (24)
Therefore for ∈ ∈ , the conolution product | ＊ | y belongs to F n:1 , if ∈ ∈ || | | | | 2 2 | |2 ∑ | 1 | || | | · |∑ | | || | | (25)
Now, we show that the analytic Wiener integral of the convolution of two first variations can be successfully expressed as the product of two analytic Wiener integrals of each first variation on the abstract Wiener space.
Theorem 3.4. For ∈ and for ∈ ,
| ＊ | dm(x) | ∙ | 26 Proof. First we have that for all real 0, | ＊ | dm(x) = | ＊ | dm(x) , ~ , ⋯ , , ~ ∑ d ∑ , ~ √ ∑ , ~∙ √ ∑ d 2 d ∑ , ~ , ⋯ , 1 2 , ~ 1, ⋯ , 2 2 2 1 d d ∑ , ~ ∑ d . ∙ ∑ , ~ ∑ d | ∙ | 27
, where we use the transformation :
√ ,
√
and , , . . . , and , , . .. , . By the
analytic continuation in ∈ , we have that 27 holds for every ∈
Finally, we show that the analytic Feynman integral of the convolution of two first variation can be successfully expressed as the product of two analytic Feynman integrals in the abstract Wiener space.
Theorem 3.5. Let F and G be functions of the form (9). Then we have that for ∈ and for ∈ ,
| ＊ | dm(x)
| ∙ | 28
Proof. By Theorem 3.4 and by the definition of the analytic Feynman integral, we have that | ＊ | dm(x) lim → | ＊ | dm(x) = lim → | ∙ lim→ | | ∙ | 29 , whenever → through ACKNOWLEDGEMENT
This work is supported by a National Research Foundation : NRF 2017R1A6A3A11030667 .
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