Parts formulas involving the Fourier–Feynman transform associated with Gaussian paths 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 ( α 1 , x , . . . , α n , x ) for scale almost every x ∈ C 0 [0, T ], where α, x denotes the Paley-Wiener-Zygmund stochastic integral T 0 α(t)dx(t), and {α 1 , . . . , α n } is an orthogonal set of nonzero functions in L 2 [0, T ]. The Gaussian processes used in this paper are not stationary.