# Parts formulas involving the Fourier-Feynman transform associated with Gaussian process on Wiener space

## TL;DR

This paper develops integration by parts formulas involving Fourier-Feynman transforms and Gaussian processes on Wiener space, extending the mathematical tools for analyzing functionals of non-stationary Gaussian processes.

## Contribution

It establishes new formulas on Wiener space using a general Cameron--Storvick theorem, applicable to non-stationary Gaussian processes and functionals with specific integral forms.

## Key findings

- Derived integration by parts formulas for generalized analytic Feynman integrals.
- Extended Fourier-Feynman transform techniques to non-stationary Gaussian processes.
- Provided a framework for analyzing functionals involving Paley--Wiener--Zygmund integrals.

## Abstract

In this paper, using a very general Cameron--Storvick theorem on the Wiener space $C_0[0,T]$, we establish various integration by parts formulas involving generalized analytic Feynman integrals, generalized analytic Fourier--Feynman transforms, and the first variation (associated with Gaussian processes) of functionals $F$ on $C_0[0,T]$ having the form $F(x)=f(\langle{\alpha_1,x}\rangle, \ldots, \langle{\alpha_n,x}\rangle)$ for scale almost every $x\in C_0[0,T]$, where $\langle{\alpha,x}\rangle$ denotes the Paley--Wiener--Zygmund stochastic integral $\int_0^T \alpha(t)dx(t)$, and $\{\alpha_1,\ldots,\alpha_n\}$ is an orthogonal set of nonzero functions in $L_2[0,T]$. The Gaussian processes used in this paper are not stationary.