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

• Published in: Banach journal of mathematical analysis 2020-04, Vol.14 (2), p.503-523
• Main Authors: Chang, Seung Jun, Choi, Jae Gil
• Format: Article
• Language: English
• Subjects: Functional Analysis; Mathematics; Mathematics and Statistics; Operator Theory; Original Paper
• ISSN: 2662-2033; 1735-8787
• DOI: 10.1007/s43037-019-00005-5

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 F ( x ) = f ( ⟨ α 1 , x ⟩ , … , ⟨ α n , x ⟩ ) for scale-invariant almost every x ∈ C 0 [ 0 , T ] , where ⟨ α , x ⟩ denotes the Paley–Wiener–Zygmund stochastic integral ∫ 0 T α ( t ) d x ( 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.