Stability of traveling wave solutions to reaction–diffusion equations driven by additive noise with Hölder continuous paths

In this paper, we investigate stability of traveling wave solutions to a class of reaction–diffusion equations perturbed by infinite-dimensional additive noise with Hölder 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 traveling wave fronts. These bounds, in terms of Hurst index and Hölder exponent, apply to a large class of infinite-dimensional self-similar drivers with Hölder continuous paths, such as linear fractional stable motion. We find that for short times, higher Hurst indices imply higher stability, while for large times, higher Hölder exponents imply higher stability.