Stochastic homogenization for a diffusion-reaction model

TL;DR

This paper advances stochastic homogenization theory for coupled diffusion-reaction systems influenced by media properties evolving via stochastic differential equations, providing deterministic limiting equations useful for porous media modeling.

Contribution

It introduces a novel homogenization framework for media properties governed by stochastic differential equations, deriving deterministic limit equations with nonlinear reactions.

Findings

Derived well-posedness of the nonlinear fine-grid problem

Formulated cell problems and deterministic limiting equations

Connected invariant measures of SDEs to homogenized models

Abstract

In this paper, we study stochastic homogenization of a coupled diffusion-reaction system. The diffusion-reaction system is coupled to stochastic differential equations, which govern the changes in the media properties. Though homogenization with changing media properties has been studied in previous findings, there is little research on homogenization when the media properties change due to stochastic differential equations. Such processes occur in many applications, where the changes in media properties are due to particle deposition. In the paper, we investigate the well-posedness of the nonlinear fine-grid (resolved) problem and derive limiting equations. We formulate the cell problems and derive the limiting equations, which are deterministic with nonlinear reaction terms. The limiting equations involve the invariant measures corresponding to stochastic differential equations. These obtained results can play an important role for modeling in porous media and allow the use of simplified and deterministic limiting equations.