Algebraic structure of the $L_2$ analytic Fourier-Feynman transform associated with Gaussian processes on Wiener space

TL;DR

This paper explores the algebraic properties of the $L_2$ analytic Fourier-Feynman transforms on Wiener space, revealing their structure as group isomorphisms of linear operators linked to Gaussian processes.

Contribution

It develops rotation properties of generalized Wiener integrals and demonstrates that these transforms form a group of linear operator isomorphisms on Hilbert space.

Findings

Transforms are linear operator isomorphisms.

The class of transforms forms a group isomorphic structure.

Rotation properties of Wiener integrals are established.

Abstract

In this paper we study algebraic structures of the classes of the $L_2$ analytic Fourier-Feynman transforms on Wiener space. To do this we first develop several rotation properties of the generalized Wiener integral associated with Gaussian processes. We then proceed to analyze the $L_2$ analytic Fourier-Feynman transforms associated with Gaussian processes. Our results show that these $L_2$ analytic Fourier--Feynman transforms are actually linear operator isomorphisms from a Hilbert space into itself. We finally investigate the algebraic structures of these classes of the transforms on Wiener space, and show that they indeed are group isomorphic.