A Langevin-Type Stochastic Differential Equation on a Space of Generalized Functionals

Abstract
Recently, Deuschel has obtained a fluctuation result for system of lattice valued diffusion processes. The result obtained is similar to the ones for mean-field interacting particle diffusions treated in a number of papers. In another direction, Kallianpur and Wolpert have introduced a class of stochastic differential equations (SDE's) governing nuclear space valued processes as model for voltage potentials for spatially extended neurons. This paper is motivated by both the above problems, especially, the problem of interacting systems. The techniques developed in this paper enable us to prove a general result which yields a central limit theorem for such systems. It also provides another approach to the fluctuation theorem in another document. In addition, the identification problem of limit measures leads us to discuss the uniqueness of weak solutions of the SDE.

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