The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation

TL;DR

This paper analyzes the long-term behavior of stochastic reaction-diffusion equations, showing that solutions evolve with a noise-perturbed interface driven by curvature, without requiring front regularity.

Contribution

It introduces a global-in-time analysis of stochastic Allen-Cahn equations using novel stochastic solution concepts and generalized front propagation theory.

Findings

Long-time behavior governed by curvature-dependent interface with additive noise

No regularity assumptions needed on the evolving front

Development of stochastic solution framework for degenerate parabolic equations

Abstract

We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i)~the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii)~the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.