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

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.