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

Abstract

We investigate the stability of traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a stochastic partial differential equation (SPDE) coupled to an ordinary differential equation (ODE). This approach has been employed by Krüger and Stannat [46] 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 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. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs that are not self-adjoint. This provides a major generalization as these systems are prevalent in many applications.

Quasi-Ergodicity of Transient Patterns in Stochastic Reaction-Diffusion Equations

“…Originally, the goal of this study was to obtain a general theory for understanding pulse-type travelling waves in stochastic reaction-diffusion equations of the form (3) and neural field equations. Previous studies of this topic include Adams & MacLaurin [2], Bresslof & Weber [10], Eichinger, Gnann, & Kuehn [28], Hamster & Hupkes [35], [36], Inglis & MacLaurin [37], Krüger & Stannat [40], [41], Lang [44], Lang & Stannat [45], MacLaurin [47], and Stannat [59]. Additionally, there is the well-documented case of front-type stochastic travelling waves in FKPP type equations: See for instance the landmark papers of Brunet & Derrida [11], [12], Mueller, Mytnik, & Quastel [49], Mueller, Mytnik, & Rhyzik [50], or the review article of Kuehn [42] and references therein.…”

“…If Assumption (a) could be relaxed to allow for ω = 0, then this same argument would provide us with a qualitative formula, (28), for the quasi-asymptotic speed of pulse type travelling waves on one-dimensional, periodic spatial domains, perturbed by additive noise. Indeed, on such spatial domains, pulse type travelling waves are simply periodic solutions of the PDE (1).…”

An Itô Formula for Isochron Maps in Separable Hilbert Space

“…The solution theory of (20) can be worked out following [1], with some modifications. However, to the author's knowledge, the solution theory for (20) when D has some zero diagonal entries has only been worked out in the setting of trace class noise [2]. In this section we simply assume the existence and uniqueness of mild solutions to (20) with nontrace class noise.…”

The asymptotic frequency of stochastic oscillators

“…in [23], [24]. It could also provide a theoretical framework for other works on the effects of noise on the speed of travelling waves, such as [16], [30].…”