Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations

TL;DR

This paper studies the stability of traveling pulses in stochastic FitzHugh-Nagumo equations with additive noise, introducing a novel approach to handle non-self-adjoint linearizations with essential spectrum parallel to the imaginary axis.

Contribution

It adapts spectral projection methods using Riesz projections for analyzing traveling wave stability in coupled SPDE-ODE systems with non-self-adjoint linearizations.

Findings

Demonstrates stability analysis method for stochastic traveling pulses

Extends spectral projection techniques to non-self-adjoint operators

Applicable to a broad class of SPDE systems with similar spectral properties

Abstract

We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Kr\"uger and Stannat for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.