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

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.