Stability of travelling wave solutions to reaction-diffusion equations driven by additive noise with H\"older continuous paths

Abstract

In this paper we investigate stability of travelling wave solutions to a class of reaction-diffusion equations perturbed by infinite-dimensional additive noise with H\"older continuous paths, covering in particular fractional Brownian motion with general Hurst index. We obtain long- and short time asymptotic error bounds on the maximal distance from the solution of the stochastic reaction-diffusion equation to the orbit of travelling wave fronts. These bounds, in terms of Hurst index and H\"older exponent, apply to a large class of infinite-dimensional self-similar drivers with H\"older continuous paths, such as linear fractional stable motion. We find that for short times, higher Hurst indices imply higher stability, while for large times, a smaller gap between Hurst index and H\"older exponent implies stability for larger noise amplitudes.