Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

Abstract

In the theory of the analytic Feynman integral, the integrand is a functional of the standard Brownian motion process. In this note, we present an example of a bounded functional which is not Feynman integrable. The bounded functionals discussed in this note are defined in sample paths of the generalized Brownian motion process.Mathematics Subject Classification. 28C20, 60J65, 46G12.1. Introduction. The purpose of this note is to illustrate an effect of drift of the generalized Brownian motion process (GBMP). To do this, we discuss the theory of analytic Feynman integrals. Frankly speaking, in order to emphasize an effect of drift of GBMPs, we present an example of a bounded functional which is not analytic Feynman integrable on the function space C a,b [0, T ]. The function space C a,b [0, T ] is a probability space induced by a GBMP. Let W ≡ C 0 [0, T ] denote one-parameter Wiener space; this is the space of all real-valued continuous functions x on [0, T ] with x(0) = 0. Let M denote the class of all Wiener measurable subsets of C 0 [0, T ] and let m be the Wiener measure. Then, as is well known, (C 0 [0, T ], M, m) is a complete measure space. The coordinate process W on C 0 [0, T ] × [0, T ] defined by (x, t)

Analytic operator-valued generalized Feynman integral on function space

“…Thus a GBMP is determined by the continuous functions a(t) and b(t). The function space C a,b [0, T ], induced by GBMP, was introduced by Yeh [26,27] and was used extensively in [6,7,8,9,10,11,12,13,14]. The function space C a,b [0, T ] used in [6,7,8,9,10,11,12,13,14] can be considered as the space of sample paths of the GBMP.…”

A Banach Algebra Similar to Cameron-Storvick’s One with Its Equivalent Spaces

“…The (generalized) Paley-Wiener-Zygmund (PWZ) stochastic integrals on the (generalized) Wiener space have been used in various papers, in particular, concerning Feynman integration theories [1][2][3][4]. In particular, the PWZ integral is used in the definition of Cameron-Storvick's Banach algebra S of functions on 0 [0, ] which is the space of generalized Fourier-Stieltjes transforms of the C-valued and finite Borel measures on 2 [0, ] [1]. Johnson [4] showed that S is isometrically isomorphic to the Banach algebra of the Fresnel integrable functions given by Albeverio and Høegh-Krohn [5].…”

Generalized transforms and generalized convolution products associated with Gaussian paths on function space

“…On the other hand, in [5,6,8], the authors introduced the FFTs on the very general function space C a,b [0, T ] (rather than the Wiener space C 0 [0, T ]), and studied their properties and related topics. The function space C a,b [0, T ], induced by a generalized Brownian motion process (GBMP), was introduced by Yeh [22,23] and was used extensively in [4,5,6,7,8,9].…”

“…In particular, the relations such as (1.2) and (1.4) do not hold between the GFFT and the GCP on the function space C a,b [0, T ]. For some previous work on the GFFT and the GCP, see [7], and for a more detailed study of the effect of drift on the GBMP, see [4] and the references cited therein.…”

“…The Wiener process used in [1,2,3,12,13,14,16,20] is stationary in time and is free of drift, while the Gaussian process used in [10,11,15,19] is non-stationary in time and is free of drift. However the stochastic processes used in this paper, as well as in [4,5,6,7,8,9], are non-stationary in time and are subject to a drift a(t), and can be used to explain the position of the Ornstein-Uhlenbeck process in an external force field [17]. But We then complete this function space to obtain the complete probability measure space…”