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 $F(x)=f(⟨{\alpha }_1,x⟩,\dots ,⟨{\alpha }_n,x⟩)$ for scale almost every $x\in C_0[0,T]$, where $⟨\alpha ,x⟩$ denotes the Paley--Wiener--Zygmund stochastic integral ${\int }_0^T\alpha (t)dx(t)$, and $\{{\alpha }_1,\dots ,{\alpha }_n\}$ is an orthogonal set of nonzero functions in $L_2[0,T]$. The Gaussian processes used in this paper are not stationary.