In  , M. D. Brue introduced the functional transform on the Feynman integral (1972). In  , R. H. Cameron wrote the paper about the translation pathology of a Wiener space (1954). In    , R. H. Cameron and W. T. Martin proved some theorems on the transformation and the translation and used the expression of the change of scale for Wiener integrals (1944-1947). In  and  , R. H. Cameron and D. A. Storvick, proved relationships between Wiener integrals and analytic Feynman integrals to prove a change of scale formula for Wiener integrals (1987). In  and  , properties among the schrödinger operator and the Wiener Integral and the Feynman integral and the Feynman’s operational calculus were studied. In  , G. W. Johnson and D. L. Skoug proved a scale-invariant measurability on the Wiener space (1979).
In  and  , Y. S. Kim proved relationships between Wiener integrals and analytic Feynman integrals and proved a change of scale formula for Wiener integrals about cylinder functions on the abstract Wiener space (1998-2001). In     , Kim proved relationships among the Fourier transform and the Fourier Feynman transform and the convolution on the abstract Wiener space (2006-2016).
In this paper, we define the scale factor for the Wiener integral and we investigate the behavior of Wiener integrals along the curve C of a scale factor
about complex valued measurable functions defined on the Wiener space , where is a Fourier-Stieltjes transform of a complex Borel measure . And we will find a very interesting behavior of a scale factor for the Wiener integral.,
2. Definitions and Preliminaries
A collection of subsets of a set X is said to be a σ-algebra in X if has the following properties: 1) , 2) If , then , (where At is the complement of A relative X), 3) If and for , then . If is a σ-algebra in X, then X is called a measurable space and the members of are called the measurable set in X. If X is a measurable space and Y is a topological space and f is a mapping of X into Y, then f is a Lebesgue-measurable function, or more briefly, a measurable function, provided that is a measurable set in X for every open set V in Y.
Let denote the space of real-valued continuous functions x on such that . Let denote the class of all Wiener measurable subsets of and let m denote Wiener measure and be a Wiener measure space and we denote the Wiener integral of a functional F by . A subset E of is said to be scale-invariant measurable if for each , and a scale-invariant measurable set N 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 F and G are equal s-a.e., we write (for more details, see  ).
Throughout this paper, let denote the n-dimensional Euclidean space and let , and 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 F 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 F over with parameter z, and for each , we write
Let q be a non-zero real number and let F be a function on whose analytic Wiener integral exists for each z in . If the following limit exists, then we call it the analytic Feynman integral of F over with parameter q, and we write
where z approaches through and .,
Now we introduce the following Wiener Integration Formula.
Theorem 2.2. Let be a Wiener space and let . Then
where is a Lebesgue measurable function and and .
In the next section, we will use the following integration formula:
where a is a complex number with , b is a real number, and .
3. Behavior of a Scale Factor for the Wiener Integral
We investigate the behavior of the scale factor for the function space integral for functions
Definition 3.1. Let be defined by
which is a Fourier-Stieltjes transform of a complex Borel measure with , where is a set of complex Borel measures defined on R.,
Remark. If we define a function on R by , then the Fourier-Stieltzes transform has some properties that 1) for all , and , where denotes the conjugate complex of . 2) f is uniform continuous in R. To see this, we write for all u and h,
, where the last integrand is bounded
by 2 and tends to 0 as for each and the last integral is bounded by . Hence the integral converges to 0 by the bounded convergence theorem. Since it does not involve , the convergence is uniform with respect to .,
Notation. Let be defined by
To expand the main result of this paper and to apply the Wiener integration formula and to prove the existence of the Wiener integral of in (6), we need to express F(x) as the function of the form .
Lemma 3.2. Let be defined by (6) and (7). Then we have that
where is a countably additive Borel measure defined on for each .
Proof. Using the series expansion of the exponential function, we have that
where and is a complex Borel measure defined on R and for each and .,
Remark. For more details about properties of the function in (6) and (7), see the chapter 15 of the book  . Some properties of the exponential function of  give me a good motivation about this paper. Especially, the third equality in (10) follows from the Equation (15.3.17) in  .,
Theorem 3.3. For and for each and for functions in (6) and for real , the Wiener integral exists and is of the form:
where is a countably additive complex Borel measure defined on for each and .
Proof. By the Wiener integration formula, we have that for real ,
where . The last equality in (12) can be proved by the mathematical induction.,
By the above result, we can investigate a very interesting behavior of the Wiener integral.
Definition 3.4. We define the scale factor for the Wiener integral by the varying real number such that
where is a complex valued function defined on R.
Property 3.1. Behavior of the scale factor for the Wiener Integral.
We investigate the interesting behavior of the scale factor for the Wiener integral by analyzing the analytic Wiener integral as followings: For real ,
Example 1. For the scale factor , we can investigate the very interesting behavior of the Wiener integral:
1) We can investigate the behavior of the Wiener integral as the varying scale factor by re-interpreting the analytic Wiener integral!
2) The exponential term of the Wiener integral is decreasing, whenever the scale factor is increasing. The exponential term of the Wiener integral is increasing, whenever the scale factor is decreasing.
3) The function is a decreasing function of , because the exponential function is a decreasing function of .
That is, the absolute value of the Wiener integral is a decreasing function about the scale factor and
Conclusion. What we have done in this research is that we first define the scale factor for the Wiener integral and later, we investigate the very interesting behavior of the scale factor for the Wiener integral. From these results, we find a new property for the Wiener integral as a function of a scale factor!
Remark. The solution of the heat equation , is
where and and is a Rd-valued continuous function defined on such that and E denotes the expectation with respect to the Wiener path starting at time and is the energy operator(or, Hamiltonian) and Δ is a Laplacian and is a potential. This formula is called the Feynman-Kac formula. For more details, see the paper  and the book  .,
Research fund of this paper is supported by NRF-2017R1A6A3A11030667 as a research professor in the project of a National Research Foundation.