Soft and hard wall in a stochastic reaction diffusion equation

TL;DR

This paper studies the asymptotic behavior of phase fronts in a stochastic reaction-diffusion system with boundary conditions, revealing how noise intensity and domain size influence front dynamics and boundary repulsion effects.

Contribution

It introduces a scaling limit analysis showing the front's evolution as a diffusion process with soft and hard boundary repulsions in a stochastic setting.

Findings

Front evolves as a diffusion process with non-linear drift

Soft boundary repulsion modeled by a non-linear drift term

Hard boundary repulsion achieved through additional diffusive scaling

Abstract

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a "soft" repulsion from the boundary. We finally show how a "hard" repulsion can be obtained by an extra diffusive scaling.