Effective macroscopic dynamics of stochastic partial differential equations in perforated domains

Abstract

An effective macroscopic model for a stochastic microscopic system is derived. The original microscopic system is modeled by a stochastic partial differential equation (SPDE) defined on a domain perforated with small holes or heterogeneities. The homogenized effective model is still an SPDE but defined on a unified domain without holes. The solutions of the microscopic model are shown to converge to those of the effective macroscopic model in probability distribution as the size of holes diminishes to zero. Moreover, the long time effectivity of the macroscopic system, in the sense of convergence in probability distribution, and the effectivity of the macroscopic system, in the sense of convergence in energy, are also proved. © 2007 Society for Industrial and Applied Mathematics.