Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems

TL;DR

This paper proves the long-term stability of planar travelling waves in multidimensional stochastic reaction-diffusion systems with multiplicative noise, using advanced stochastic calculus techniques to handle anticipative integrals.

Contribution

It introduces a stochastic phase tracking method for multidimensional stability analysis of reaction-diffusion waves under noise, extending the theory of forward integrals to this context.

Findings

Stability of planar waves persists over exponentially long time scales.

Development of a stochastic phase tracking mechanism for noisy PDEs.

Extension of forward integral theory to handle anticipative stochastic integrals.

Abstract

We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations. In the deterministic setting, multidimensional stability of planar waves on the whole space $\mathbb R^d$ has been studied by many. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. This is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of It\^o-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.