Conditional Fourier-Feynman Transforms with Drift on a Function Space

Dong Hyun Cho,Suk Bong Park

doi:10.1155/2019/9483724

Abstract

In this paper we derive change of scale formulas for conditional analytic Fourier-Feynman transforms and conditional convolution products of the functions which are the products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the complex Borel measures on L2[0,T] using two simple formulas for conditional expectations with a drift on an analogue of Wiener space. Then 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 establish various changes of scale formulas for the conditional transforms and the conditional convolution products.

Highlights

Let C0[0, T] denote the Wiener space, the space of realvalued continuous functions x on the interval [0, T] with x(0) = 0
Yoo and his coauthors [3] presented a change of scale formula for Wiener integrals of functions on the abstract Wiener space B [4] which generalizes C0[0, T]
We note that the functions used in [3] are the products of generalized cylinder functions on B and the functions on the Fresnel class [5] which is the space of Fourier-Stieltjes transforms of measures on a separable real Hilbert space densely embedded in B, and note that they need not be bounded or continuous

Summary

Introduction

Denotes the Paley-Wiener-Zygmund stochastic integral [6], and h(≠ 0 a.e.) and a are of bounded variation and absolutely continuous, respectively, on [0, T] He used the conditioning function Zn+1(x) = (Z(x, t0), Z(x, t1), . In this paper, using two simple formulas for conditional expectations over paths [10, 11], we evaluate conditional expectations of the products of generalized cylinder functions and the functions in a Banach algebra which plays significant roles in Feynman integration theories and quantum mechanics. We note that the functions in (1) extend the initial state of the Schrodinger equation [12]

A Function Space and Preliminary Results

Generalized Conditional Fourier-Feynman Transforms

Generalized Conditional Convolution Products

Relationships between Transforms and Convolution Products

Change of Scale Formulas for the Transforms and Convolutions