Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment

TL;DR

This paper develops a sigma-convergence method for homogenizing nonlinear stochastic PDEs in ergodic environments, demonstrating convergence of solutions and applying results to various physical scenarios.

Contribution

It introduces a sigma-convergence approach for stochastic PDE homogenization, extending the method to nonlinear equations in ergodic settings.

Findings

Solutions converge in probability to a homogenized equation.

The method applies to periodic, almost periodic, and weakly almost periodic environments.

Homogenization results are validated for concrete physical models.

Abstract

In this paper, 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 problem for a stochastic nonlinear partial differential equation is studied. 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 a suitable version of sigma-convergence method, the sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem. We apply the homogenization result to some concrete physical situations such as the periodicity, the almost periodicity, the weak almost periodicity, and others.