Stability of Travelling Waves on Exponentially Long Timescales in Stochastic Reaction-Diffusion Equations

TL;DR

This paper proves that travelling waves in certain stochastic reaction-diffusion equations remain stable over exponentially long timescales by extending phase-tracking techniques and employing advanced probabilistic bounds.

Contribution

It extends phase-tracking methods to exponentially long timescales in stochastic PDEs using generic chaining and stochastic convolution bounds.

Findings

Travelling waves exhibit meta-stability over exponentially long times.

Phase-tracking techniques are effective in stochastic reaction-diffusion equations.

Logarithmic supremum bounds are established for stochastic convolutions.

Abstract

In this paper we establish the meta-stability of travelling waves for a class of reaction-diffusion equations forced by a multiplicative noise term. In particular, we show that the phase-tracking technique developed in [hamster2017,hamster2020] can be maintained over timescales that are exponentially long with respect to the noise intensity. This is achieved by combining the generic chaining principle with a mild version of the Burkholder-Davis-Gundy inequality to establish logarithmic supremum bounds for stochastic convolutions in the critical regularity regime.