Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

Abstract

In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form F ( x ) = f ( ⟨ α 1 , x ⟩ , … , ⟨ α n , x ⟩ ) F(x)=f(\langle \alpha _{1} , x\rangle , \dots , \langle \alpha _{n} , x\rangle ) where ⟨ α , x ⟩ \langle {\alpha ,x}\rangle denotes the Paley-Wiener-Zygmund stochastic integral ∫ 0 T α ( t ) d x ( t ) \int _{0}^{T} \alpha (t) d x(t) .