Generalized Analytic Feynman Integral via Function Space Integral of Bounded Cylinder Functionals

Abstract

Abstract. In this paper, we use a generalized Brownian motion to define a generalized analytic Feynman integral. We then obtain some results for the generalized analytic Feynman integral of bounded cylinder functionals of the formdefined on a very general function space C a,b [0, T ]. We also present a change of scale formula for function space integrals of such cylinder functionals.

A Fubini theorem on a function space and its applications

“…For a detailed study of the theories, see [18,Section 5]. Also, see [4,5,9,11,13,14,15,16,17,20] for related work.…”

“…The function space C a,b [0, T ], induced by generalized Brownian motion, was introduced by Yeh in [22] and was used extensively in [4,5,6,7,8,9].…”

“…The Wiener process used in [1,2,3,10,11,12,13,14,15,16,17,20] is free of drift and is stationary in time while the stochastic process used in this paper, as well as in [4,5,6,7,8,9], is nonstationary in time, is subject to a drift a(t), and can be used to explain the position of the Ornstein-Uhlenbeck process in an external force field [19]. Thus the formulas and results in this paper are more complicated than the formulas and results in [1,3].…”

Generalized Fourier-Feynman Transform and Sequential Transforms on Function Space

“…In this paper, as in [3], [4], [5], [6], [8], [9], we consider the incomplete function space (C a,b [0, T ], B(C a,b [0, T ]), µ) and we denote the function space integral of a B(C a,b [0, T ])-measurable functional F by…”

“…The Wiener process used in [1], [2], [10], [11], [12], [14] is free of drift and stationary in time while the stochastic process used in this paper, as well as in [3], [4], [5], [6], [7], [8], [9] is nonstationary in time and is subject to a drift a(t).…”