Complexity
Volume 2020 (2020), Article ID 2671474, 7 pages
https://doi.org/10.1155/2020/2671474

Research Article

Analytic Feynman Integral and a Change of Scale Formula for Wiener Integrals of an Unbounded Cylinder Function

Kim Young Sik

Department of Mathematics, College of Natural Sciences, Industry-University Cooperation Foundation, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Republic of Korea

Correspondence should be addressed to Kim Young Sik; yoskim@hanyang.ac.kr

Received 18 June 2020; Revised 14 September 2020; Accepted 22 September 2020; Published 17 October 2020

Academic Editor: Dimitri Volchenkov

Copyright © 2020 Kim Young Sik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate the behavior of the unbounded cylinder function whose analytic Wiener integral and analytic Feynman integral exist, we prove some relationships among the analytic Wiener integral, the analytic Feynman integral, and the Wiener integral, and we prove a change of scale formula for the Wiener integral about the unbounded function on the Wiener space .

1. Introduction

In [1], Cameron and Martin initially worked about the behavior of measure and measurability under change of scale in the Wiener space in 1947. In [2], Johnson and Skoug proved the scale-invariant measurability on the Wiener space in 1979. In [3, 4], Cameron and Storvick proved a change of scale formula for bounded functions on the Wiener space in 1987. In [5], Kim proved a change of scale formula for Wiener integrals about a function with : the analytic Wiener integral exists for and the analytic Feynman integral exists for . In general, the analytic Feynman integral does not always exist for with . In [6], Brue worked about the transform for Feynman integrals in 1972. In [7], Huffman et al. expanded the Fourier Feynman transform theory of for with . In [8, 9], Kim extended these results to the function , where is a Fourier transform of a complex-valued Borel measure in , which is a space of complex-valued Borel measures. In [10], Kim investigated the behavior of a scale factor for Wiener integrals on the Wiener space.

In [11]‐[13], Cameron and Martin expanded the theory about the translation and transformation theory for the Wiener integral. In [14], Chung expanded the generalized integral transforms for Wiener integrals. In [15], Gaysinsky and Goldstein expanded the self-adjointness of Schrodinger operator and Wiener integrals. In [16], Johnson and Lapidus wrote the book about the Feynman integral and the Feynman's operational calculus. In [20], Kim proved the change of scale formula for Wiener integrals of cylinder functions of a Fourier transform of a measure.

In this paper, we investigate the behavior of a Wiener integral for the unbounded function and we prove that is Wiener integrable and the analytic Wiener integral and the analytic Feynman integral of exist. We also prove some relationships among the analytic Wiener integral, the analytic Feynman integral, and the Wiener integral and prove a change of scale formula for the Wiener integral of the unbounded function on the Wiener space .

2. Definitions and Preliminaries

Throughout this paper, let denote the -dimensional Euclidean space and let , and denote the set of complex numbers, the set of complex numbers with positive real part, and the set of nonzero complex numbers with nonnegative real part, respectively.

Let denote the space of real-valued continuous functions on such that . Let denote the class of all Wiener measurable subsets of  , let denote a Wiener measure, and let be a Wiener measure space; we denote the Wiener integral of a function by .

A subset of is said to be scale-invariant measurable if for each , and a scale-invariant measurable set is said to be scale-invariant null if for each . A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). If two functionals and are equal s-a.e., we write .

Definition 1. Let be a complex-valued measurable function on such that the integral

exists for all real . If there exists a function analytic on such that for all real , then we define to be the analytic Wiener integral of over with parameter , and for each , we write

Let be a nonzero real number and let be a function defined on whose analytic Wiener integral exists for each in . If the following limit exists, then we call it the analytic Feynman integral of over with parameter , and we write

where approaches through and .

Theorem 1. Wiener integration formula:

where is an orthonormal set of elements in , is a Lebesgue measurable function, , and .

Remark 1. We will use several times the following well-known integration formula:

where is a complex number with and is a real number.

3. Main Results

Define the unbounded function :

where is an orthonormal set in and , and is unbounded.

To expand main results of this paper, we prove some lemmas.

Lemma 1. For and for , we have that

Proof. First we know that for and for ,

Using the series expansion of the exponential function   with , we have that for and for ,

And for ,

By (8)–(10), we have the desired result.

Lemma 2. Let be the unbounded function defined by (6). Then, for , is a Wiener integrable function of and the Wiener integral is

Proof. By (4) and by Lemma 1, we have that

Remark 2. By Lemma 2, we have interesting Wiener integrals about the unbounded function: for an orthonormal set in ,

Because is a Wiener integrable function even though it is unbounded, we can challenge to prove the change of scale formula for the Wiener integral about the unbounded function in (6) on the Wiener space .
First, we prove the existence of the analytic Wiener integral and the analytic Feynman integral about the unbounded function in (6) on the Wiener space .

Theorem 2. Let be the unbounded function defined by (6). Then, for and for , the analytic Wiener integral and the analytic Feynman integral of exist and are given by

whenever through .

Proof. By (4) and by Lemma 2, we have that for real and for ,

By the analytic continuation of , we can deduce the desired analytic Wiener integral and the analytic Feynman integral of on the Wiener space .

Remark 3. In Theorem 2, we prove that the analytic Wiener integral and the analytic Feynman integral about the unbounded function can exist, even though , , on the Wiener space .
To investigate the behavior of a change of scale formula for the Wiener integral, we prove some relationships between the Wiener integral and the analytic Wiener integral about the unbounded function in (6) on the Wiener space .

Lemma 3. Let be the unbounded function defined by (6). For ,

is a Wiener integrable function of .

Proof. By (4) and by (7), we have that for and for ,

Therefore, we have the desired result.
We prove the relationship between the analytic Wiener integral and the Wiener integral for the unbounded function in (6). That is, we prove that the analytic Wiener integral of can be successfully expressed as the sequence of Wiener integrals on the Wiener space .

Theorem 3. Let be the unbounded function defined by (6). Then, for , the analytic Wiener integral of can be successfully expressed as the sequence of Wiener integrals:

Proof. By the proof of Lemma 3, we have that for and for ,

We prove that the unbounded function in (6) successfully satisfies the change of scale formula for the Wiener integral on the Wiener space .

Theorem 4. Let be the unbounded function defined by (6). Then, for a positive real ,

Proof. By Theorem 3, we have that for real ,

If we let in the above equation, we have the desired result.
Finally, we prove the relationship between the analytic Feynman integral and the Wiener integral. That is, we prove that the analytic Feynman integral of the unbounded function can be successfully expressed as the limit of the sequence of Wiener integrals on the Wiener space .

Theorem 5. Let be the unbounded function defined by (6). Then, the analytic Feynman integral of can be successfully expressed as the limit of the sequence of analytic Wiener integrals:

whenever through with .

Proof. By Theorem 3 and by the definition of the analytic Feynman integral, we have that

whenever through .

Remark 4. The motivation of this paper follows from the notation and by some properties on the Hilbert space in [18, 19]. To check the existence of the analytic Wiener integral and the analytic Feynman integral of , we take and there are no other reasons about this choice.

Data Availability

The data used to support the findings of this study are included within this article.

Disclosure

The abstract of this paper was presented in the international conference of Korean Mathematical Society in 2017 [20].

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study was supported by the National Research Foundation of Korea (grant no. NRF-2017R1A6A3A11030667).

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