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

TL;DR

This paper studies the stability of travelling wave solutions in reaction-diffusion equations affected by infinite-dimensional additive noise with H"older continuous paths, providing bounds that depend on the Hurst index and H"older exponent.

Contribution

It introduces a framework for analyzing stability under a broad class of H"older continuous noise, including fractional Brownian motion, with explicit error bounds.

Findings

Higher Hurst indices increase short-term stability.

Smaller gap between Hurst index and H"older exponent enhances long-term stability.

Results apply to various self-similar infinite-dimensional noise processes.

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.