A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

Dong Hyun Cho

doi:10.1155/2017/8510782

Abstract

Using a simple formula for conditional expectations over continuous paths, we will evaluate conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the measures on the Borel class of L2[0,T]. We will then investigate their relationships. Particularly, we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we will establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and change of scale formulas, we use multivariate normal distributions so that the conditioning function does not contain present positions of the paths.

Highlights

Let C[0, T] denote an analogue of Wiener space which is the space of real-valued continuous functions on the interval [0, T] [1]
According to the author’s paper, he investigated the effects that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur among them
He derived several change of scale formulas for the conditional transforms and the conditional convolution products, which simplify evaluating the conditional expectations, because the probability measure used on C[0, T] cannot be scale-invariant [3, 4]

Summary

Introduction

Let C[0, T] denote an analogue of Wiener space which is the space of real-valued continuous functions on the interval [0, T] [1]. In this paper, using a simple formula for conditional expectations over continuous paths [6], we evaluate conditional expectations of generalized cylinder functions and the functions in a Banach algebra which plays significant roles in Feynman integration theories and quantum mechanics. With the conditioning function Zn which does not contain the present position Z(x, T) of the path Z(x, ⋅), we evaluate conditional expectations, that is, the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by (3). Compared with the results in [5, 7], the conditioning function Zn in this paper does not contain the present position Z(x, T) of Z(x, ⋅) and the effects of drift a depend on the polygonal function of a so that we can generalize the theorems in [5, 7] and the results of this paper do not depend on a particular choice of the initial distribution of the paths

Preliminaries on an Analogue of Wiener Space

Conditional Fourier-Wiener and FourierFeynman Transforms

Evaluation Formulas for the Functions in a Banach Algebra

Change of Scale Formulas for the Transforms and Convolutions