A Rotation on Wiener Space with Applications

Jae Gil Choi,Seung Jun Chang

doi:10.5402/2012/578174

Abstract

We first investigate a rotation property of Wiener measure on the product of Wiener spaces. Next, using the concept of the generalized analytic Feynman integral, we define a generalized Fourier-Feynman transform and a generalized convolution product for functionals on Wiener space. We then proceed to establish a fundamental result involving the generalized transform and the generalized convolution product.

Highlights

Let C0 0, T denote one-parameter Wiener space, that is, the space of all real-valued continuous functions x on 0, T with x 0 0
In 1, Bearman gave a significant theorem for Wiener integral on product Wiener space
Let Tθ : C02 0, T → C02 0, T be the transformation defined by Tθ w, z w, z with t t wt cos θ s dw s − sin θ s dz s, ISRN Applied Mathematics t t zt sin θ s dw s cos θ s dz s

Summary

Introduction

Let C0 0, T denote one-parameter Wiener space, that is, the space of all real-valued continuous functions x on 0, T with x 0 0. C0 0, T , M, m is a complete measure space, and we denote the Wiener integral of a Wiener integrable functional F by. In 1 , Bearman gave a significant theorem for Wiener integral on product Wiener space. As a special case of Theorem 1.1, one can obtain the following corollary. Theorem 1.3 is used to study relationships between analytic FourierFeynman transforms and convolution products of Feynman integrable functionals on Wiener space, see for instance 3–6. We apply our rotation property of Wiener measure to establish a fundamental relationship between the generalized Fourier-Feynman transform and the generalized convolution product

A Rotation on Wiener Space

Proof of the Main Theorem

Generalized Fourier-Feynman Transform and Generalized Convolution Product