Homogenization of a stochastic nonlinear reaction-diffusion equation with a large reaction term: the almost periodic framework

TL;DR

This paper studies the homogenization of a stochastic nonlinear reaction-diffusion equation with a large reaction term under an almost periodic framework, showing convergence to a stochastic convection-diffusion equation.

Contribution

It introduces a generalized sigma-convergence method for stochastic processes and derives the effective equation in an almost periodic setting, extending previous homogenization results.

Findings

Solutions converge in probability to a stochastic convection-diffusion equation.

Explicit form of the homogenized equation is derived.

Applicable to various frameworks including periodic and almost periodic cases.

Abstract

Homogenization of a stochastic nonlinear reaction-diffusion equation with a large non- linear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic non- linear convection-diffusion equation which we explicitly derive in terms of appropriated functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable generalized concept of sigma-convergence for stochastic processes.