Homogenization of Nonlinear Stochastic Partial Differential Equations in a General Ergodic Environment

Abstract

In this article, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations. In this regard, the homogenization of a stochastic nonlinear partial differential equation is addressed. Using some deep compactness results such as the Prokhorov and Skorokhod theorems, we prove that the sequence of solutions of this problem converges in probability towards the solution of an equation of the same type. To proceed with, we use the concept of sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem.