J. Phys. A: Math. Gen. 32 (1999) 4005–4026. Printed in the UK PII: S0305-4470(99)97184-6 The generalized Wiener process II: Finite systems Adri´ an A Budini† and M O C´ aceres‡§ † Centro At ´ omico Bariloche, Av. Ezequiel Bustillo Km 9.5, 8400 San Carlos de Bariloche, R´ ıo Negro, Argentina ‡ Centro At ´ omico Bariloche and Instituto Balseiro, CNEA and Universidad Nacional de Cuyo, Av. Ezequiel Bustillo Km 9.5, 8400 San Carlos de Bariloche, R´ ıo Negro, Argentina Received 2 September 1998, in final form 9 March 1999 Abstract. A class of Langevin-like equations (non-Markovian processes) are studied in the presence of non-natural boundary conditions. Exact results for all cumulants and the corresponding Kolmogorov hierarchy of distributions are given in terms of our functional approach we previously reported (1997 J. Phys. A: Math. Gen. 30 8427). The generalized Wiener processes—on finite domains—are completely characterized for reflecting and periodic boundary conditions. Some examples are given to show the behaviour of the moments and the probability distributions for different noises. The interplay between the boundary conditions and the structure of the noises is also pointed out. 1. Introduction The behaviour of systems under the effect of noise has attracted the interest of many workers for many years [1,2]. In particular, stochastic equations to model relaxation have been studied for several purposes [3,4], and by means of different approximations [5]. As is well known, when the random force is Gaussian and white, Fokker–Planck equations for the distributions are available. In general, if any other noise is utilized, an individual and particular treatment is required. This is true even with linear stochastic differential equations (SDEs). Nevertheless, general methods to characterize some non-Markovian processes can be constructed [6]. In [6] (from now on referred to as paper I) we developed a functional approach in order to characterize Langevin-like equations—with natural boundary conditions—and driven by arbitrary structures of noise. From that approach it is possible to construct the characteristic functional of the processes, from which all statistical information can be obtained, hence providing a systematic way to calculate exact properties for a large class of non-Markovian processes. We remark that in order to obtain the characteristic functional of the non-Markovian processes, it is only necessary to know the characteristic functional of the noise. Recently there has been some interest in the effects of the boundaries on finite non- Markovian diffusion systems. This problem, to our knowledge, has only been treated with dichotomous noise [7]. Therefore, the principal object of this paper is to extend our functional approach to a particular class of finite non-Markovian processes, i.e. when there exist boundary conditions (BC) on the domain of interest D and when the noise—in the corresponding SDE— is an arbitrary stochastic process (SP) ξ(t) characterized by its functional G ξ ([k(t)]). Once again we note that our approach is exact and provides the starting point to obtain, in a systematic way, higher-order cumulants and also the whole Kolmogorov hierarchy. § E-mail address: caceres@cab.cnea.gov.ar 0305-4470/99/224005+22$19.50 © 1999 IOP Publishing Ltd 4005