Parts formulas involving the Fourier–Feynman transform associated with Gaussian paths on Wiener space

Abstract

Park and Skoug established several integration by parts formulas involving analytic Feynman integrals, analytic Fourier–Feynman transforms, and the first variation of cylinder-type functionals of standard Brownian motion paths in Wiener space $C_0[0,T]$. 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

$$\begin{aligned} F(x)=f(\langle {\alpha _1,x}\rangle , \ldots , \langle {\alpha _n,x}\rangle ) \end{aligned}$$

for scale-invariant 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.