Stochastic Homogenization for Reaction-Diffusion Equations

TL;DR

This paper establishes stochastic homogenization for reaction-diffusion equations with stationary ergodic reactions, linking front speeds, Wulff shapes, and Hamilton-Jacobi equations to describe large-scale solution behavior.

Contribution

It introduces a new relation between front speeds and Wulff shapes without corners, enabling general stochastic homogenization results for reaction-diffusion equations.

Findings

Localized solutions asymptotically form Wulff-shaped regions.

Existence of front speeds in all directions is proven.

Large-scale solutions governed by a deterministic Hamilton-Jacobi equation.

Abstract

In the present paper we study stochastic homogenization for reaction-diffusion equations with stationary ergodic reactions. We first show that under suitable hypotheses, initially localized solutions to the PDE asymptotically become approximate characteristic functions of a ballistically expanding Wulff shape. The next crucial component is the proper definition of relevant front speeds and subsequent establishment of their existence. We achieve the latter by finding a new relation between the front speeds and the Wulff shape, provided the Wulff shape does not have corners. Once front speeds are proved to exist in all directions, by the above means or otherwise, we are able to obtain general stochastic homogenization results, showing that large space-time evolution of solutions to the PDE is governed by a simple deterministic Hamilton-Jacobi equation whose Hamiltonian is given by these front speeds. We primarily consider the case of non-negative reactions but we also extend our results to the more general PDE $u_{t}= F(D^2 u,\nabla u,u,x,\omega)$ as long as its solutions satisfy some basic hypotheses including positive lower and upper bounds on spreading speeds in all directions and a sub-ballistic bound on the width of the transition zone between the two equilibria of the PDE.