Analytic Fourier--Feynman transforms and convolution type operations associated with Gaussian processes on Wiener space

TL;DR

This paper introduces a new convolution type operation for functionals on Wiener space and explores its relationship with generalized analytic Fourier--Feynman transforms, showing they can be represented as products of these transforms.

Contribution

It defines a novel convolution type operation on Wiener space and establishes fundamental relationships with generalized analytic Fourier--Feynman transforms.

Findings

Convolution type operations can be expressed via Fourier--Feynman transforms.

The transforms of convolutions are represented as products of transforms.

The work unifies various convolution concepts on Wiener space.

Abstract

In this paper we introduce the concept of a convolution type operation of functionals on Wiener space. It contains several kinds of the concepts of convolution products on Wiener space, which have been studied by many authors. We then investigate fundamental relationships between generalized analytic Fourier--Feynman transforms and convolution type operations. Both of the generalized analytic Fourier--Feynman transform of the convolution type operation and the convolution type operation of the generalized analytic Fourier--Feynman transforms are represented as a product of the generalized analytic Fourier--Feynman transforms.