Infinite dimensional Langevin equation and Fokker-Planck equation

Abstract

Stochastic processes on a Hilbert space have been discussed in connection with quantum field theory, theory of partial differential equations involving random terms, filtering theory in electrical engineering and so forth, and the theory of those processes has greatly developed recently by many authors (A. B. Balakrishnan [1, 2], Yu. L. Daletskii [7], D. A. Dawson [8, 9], Z. Haba [12], R. Marcus [18], M. Yor [26]).

Eigenfunction expansions for infinite dimensional Ornstein-Uhlenbeck processes

“…As explained in [5,Sect. 2.3] and proved in [21], this is the case for Walsh's model of neuronal behavior, and hence, Walsh's conjecture 1 of [29] is true.…”

“…This example covers most of the cases where the operator A has a compact resolvent. See for example [9,10,7,11] and [21].…”

Semi-Linear Stochastic Evolution Equations

“…Linear stochastic evolution equations have been extensively studied in recent years. They occur in Dawson (1975), Miyahara (1981), Ito (1978Ito ( , 1982, Holley and Strook (1978), Kallianpur and Wolpert (1984), Ichikawa (1978), Walsh (1981, 1986), Ustunel (1982 , Kallianpur and Perez-Abreu (1987), Da Prato et al (1982a, Da Prato (1983) , and Leon (1989). Consider the linear stochastic evolution equation…”

Infinite dimensional stochastic differential equation models for spatially distributed neurons

“…For example, It6 has an excellent account in his 1981 Evanston lecture, [12]. Miyahara considers similar processes in connection with a vibrating string problem [16]; Holley and Stroock, in their treatment of a problem involving infinite particle systems also introduce these processes [10]. However, since no discussion in the literature seems to include all of the estimates we will need in Section 3, we prefer to derive them here.…”