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

Abstract

In this paper we establish the meta-stability of traveling 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 [C. H. S. Hamster and H. J. Hupkes, SIAM J. Appl. Dyn. Syst., 18, pp. 205--278; Phys. D, 401, 132233] 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.