scholarly journals A Change of Scale Formula for Wiener Integrals about the First Variation on the Product Abstract Wiener Space

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 12
Author(s):  
Young Sik Kim
Keyword(s):  
Feynman Integral ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Wiener Integral ◽  
First Variation ◽  
Change Of Scale ◽  
Variation Of A Function ◽  
The First Variation

We shall prove the existence of the Wiener integral and the analytic Wiener and Feynman integral and we obtain those relationships and later, we prove the change of scale formula for the Wiener integral about the first variation of a function defined on the product abstract Wiener space. Later, we obtain those relationships in the Fresnel class as it’s corollaries.

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

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1666 ◽  
Author(s):  
Young Sik Kim
Keyword(s):  
Feynman Integral ◽  
Wiener Integral ◽  
Stieltjes Transform ◽  
First Variation ◽  
Borel Measures ◽  
Analytic Feynman Integral ◽  
Change Of Scale ◽  
Complex Borel Measure ◽  
The First Variation ◽  
Complex Borel Measures

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x)=exp{∫0Tθ(t,x(t))dt} successfully exist under the certain condition, where θ(t,u)=∫Rexp{iuv}dσt(v) is a Fourier–Stieltjes transform of a complex Borel measure σt∈M(R) and M(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 of F(x) sucessfully holds on the Wiener space.

scholarly journals A change of scale formula for Wiener integrals of cylinder functions on abstract Wiener space

1998 ◽  
Vol 21 (1) ◽  
pp. 73-78 ◽  
Author(s):  
Young Sik Kim
Keyword(s):  
Feynman Integral ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Wiener Integral ◽  
Feynman Integrals ◽  
Analytic Feynman Integral ◽  
Change Of Scale ◽  
The Relationship

The purpose of this paper is to establish the existence of analytic Wiener and Feynman integrals for a class of certain cylinder functions which is of the form:F(x)=f((h1,x)∼,⋯,(hn,x)∼),    x∈B,on the abstract Wiener space, and to establish the relationship between the Wiener integral and the analytic Feynman integral for such cylinder functions on the abstract Wiener space. We then establish a change of scale formula for Wiener integrals of such cylinder functions on the abstract Wiener space.

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

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Kim Young Sik
Keyword(s):  
Feynman Integral ◽  
Wiener Space ◽  
Wiener Integral ◽  
Unbounded Function ◽  
Cylinder Function ◽  
Analytic Feynman Integral ◽  
Change Of Scale ◽  
Unbounded Cylinder

We investigate the behavior of the unbounded cylinder function F x = ∫ 0 T α 1 t d x t 2 k ⋅ ∫ 0 T α 2 t d x t 2 k ⋅ ⋯ ⋅ ∫ 0 T α n t d x t 2 k ,   k = 1,2 , … 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 C 0 0 , T .

scholarly journals A change of scale formula for Wiener integrals of cylinder functions on the abstract Wiener space II

2001 ◽  
Vol 25 (4) ◽  
pp. 231-237 ◽  
Author(s):  
Young Sik Kim
Keyword(s):  
Fourier Transform ◽  
Borel Measure ◽  
Feynman Integral ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Analytic Feynman Integral ◽  
Change Of Scale ◽  
The Fourier Transform ◽  
Complex Valued

We show that for certain bounded cylinder functions of the formF(x)=μˆ((h1,x)∼,...,(hn,x)∼),x∈Bwhereμˆ:ℝn→ℂis the Fourier-transform of the complex-valued Borel measureμonℬ(ℝn), the Borelσ-algebra ofℝnwith‖μ‖<∞, the analytic Feynman integral ofFexists, although the analytic Feynman integral,limz→−iqIaw(F;z)=limz→−iq(z/2π)n/2∫ℝnf(u→)exp{−(z/2)|u→|2}du→, do not always exist for bounded cylinder functionsF(x)=f((h1,x)∼,...,(hn,x)∼),x∈B. We prove a change of scale formula for Wiener integrals ofFon the abstract Wiener space.

scholarly journals Parts formulas involving conditional Feynman integrals

2002 ◽  
Vol 65 (3) ◽  
pp. 353-369 ◽  
Author(s):  
Seung Jun Chang ◽  
David Skoug
Keyword(s):  
Basic Result ◽  
Feynman Integral ◽  
Wiener Space ◽  
Integration By Parts ◽  
Feynman Integrals ◽  
First Variation ◽  
Basic Formula ◽  
Analytic Feynman Integral ◽  
The First Variation

In this paper we first obtain a basic formula for the conditional analytic Feynman integral of the first variation of a functional on Wiener space. We then apply this basic result to obtain several integration by parts formulas for conditional analytic Feynman integrals and conditional Fourier-Feynman transforms.

scholarly journals A Partial Derivative Approach to the Change of Scale Formula for the Function Space Integral

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 26
Author(s):  
Young Sik Kim
Keyword(s):  
Fourier Transform ◽  
Function Space ◽  
Partial Derivative ◽  
Directional Derivative ◽  
Feynman Integral ◽  
Wiener Integral ◽  
Vector Calculus ◽  
Change Of Scale

We investigate the partial derivative approach to the change of scale formula for the functon space integral and we investigate the vector calculus approach to the directional derivative on the function space and prove relationships among the Wiener integral and the Feynman integral about the directional derivative of a Fourier transform.

scholarly journals Change of Scale Formulas for Wiener Integrals Related to Fourier-Feynman Transform and Convolution

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Bong Jin Kim ◽  
Byoung Soo Kim ◽  
Il Yoo
Keyword(s):  
Banach Algebra ◽  
Wiener Space ◽  
Convolution Product ◽  
Abstract Wiener Space ◽  
Change Of Scale ◽  
Generalized Fresnel Class

Cameron and Storvick discovered change of scale formulas for Wiener integrals of functionals in Banach algebraSon classical Wiener space. Yoo and Skoug extended these results for functionals in the Fresnel classF(B)and in a generalized Fresnel classFA1,A2on abstract Wiener space. We express Fourier-Feynman transform and convolution product of functionals inSas limits of Wiener integrals. Moreover we obtain change of scale formulas for Wiener integrals related to Fourier-Feynman transform and convolution product of these functionals.

scholarly journals Relationships among transforms, convolutions, and first variations

1999 ◽  
Vol 22 (1) ◽  
pp. 191-204 ◽  
Author(s):  
Jeong Gyoo Kim ◽  
Jung Won Ko ◽  
Chull Park ◽  
David Skoug
Keyword(s):  
Stochastic Integral ◽  
Wiener Space ◽  
Convolution Product ◽  
First Variation ◽  
The First Variation

In this paper, we establish several interesting relationships involving the Fourier-Feynman transform, the convolution product, and the first variation for functionalsFon Wiener space of the formF(x)=f(〈α1,x〉,…,〈αn,x〉),                                                      (*)where〈αj,x〉denotes the Paley-Wiener-Zygmund stochastic integral∫0Tαj(t)dx(t).

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