Behavior of a Scale Factor for Wiener Integrals and a Fourier Stieltjes Transform on the Wiener Space

The purpose of this paper is to investigate the behavior of a Wiener integral along the curve C of the scale factor 0 ρ > for the Wiener integral [ ] ( ) ( ) 0 0, d C T F x m x ρ ∫ about the function ( ) ( ) ( ) { } 0 exp , d T F x t x t t θ = ∫ defined on the Wiener space [ ] 0 0, C T , where ( ) , t u θ is a Fourier-Stieltjes transform of a complex Borel measure.

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 0 ρ > about complex valued measurable functions ( ) ( ) Fourier-Stieltjes transform of a complex Borel measure t σ .And we will find a very interesting behavior of a scale factor 0 ρ > for the Wiener integral.

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) 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 ( )  C C , and + 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 F be a complex-valued measurable function on exists for all real 0 λ > .If there exists a function ( ) for all real 0 λ > , then we define ( ) where z approaches iq Now we introduce the following Wiener Integration Formula.Theorem 2.2.Let [ ] 0 0, C T be a Wiener space and let where In the next section, we will use the following integration formula: where a is a complex number with 0 Rea > , b is a real number, and 2 1 i = − .

Behavior of a Scale Factor for the Wiener Integral
We investigate the behavior of the scale factor for the function space integral for functions be defined by which is a Fourier-Stieltjes transform of a complex Borel measure , then the Fourier-Stieltzes transform has some properties that 1) for all u R ∈ , ( ) where z denotes the conjugate complex of z ∈C . 2)f is uniform continuous in R. To see this, we write for all u and h, 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 ( ) (6), we need to express F(x) as the function of the form T → C be defined by ( 6) and (7).Then we have that where n µ is a countably additive Borel measure defined on Proof.Using the series expansion of the exponential function, we have that ( where ( ) Remark.For more details about properties of the function ( ) 6) and (7), see the chapter 15 of the book [9].Some properties of the exponential function of [9] give me a good motivation about this paper.Especially, the third equality in (10) follows from the Equation (15.3.17) in [9].Theorem 3.
where n µ is a countably additive complex Borel measure defined on Proof.By the Wiener integration formula, we have that for real . 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 where : G R C → 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 0  ( ) [ ] ( ) ( ) Remark.<Interpretation of the scale factor for 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 0 ρ > is increasing.The exponential term of the Wiener integral is increasing, whenever the scale factor 0 ρ > is decreasing.
3) The function This formula is called the Feynman-Kac formula.For more details, see the paper [8] and the book [9].Remark.<Gratitude for the Refree> I am very gratitude for the referee to comment in details.

T
denote the space of real-valued continuous functions x on [ ] 0,T such that ( ) 0 0 x = .Let  denote the class of all Wiener measurable subsets of measure space and we denote the Wiener integral of a functional F by [ ] ( ) ( )

T
3.  For z + ∈C and for each 1, 2, , → C in(6) and for real 0 ρ > , the Wiener integral exists and is of the form:

x
is, the absolute value of the Wiener integral is a decreasing function about the scale factor 0 Conclusion.What we have done in this research is that we first define the Y. S. Kim DOI: 10.4236/am.2018.95035494 Applied Mathematics 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 U ⋅ is a R d -valued continuous function defined on [ ] 0,t such that ( ) 0 0 x = and E denotes the expectation with respect to the Wiener path starting at time 0 t = and H V = −∆ + is the energy operator(or, Hamiltonian) and Δ is a Laplacian and