Multiscale analysis for traveling-pulse solutions to the stochastic fitzhugh–nagumo equations

Multiscale analysis for traveling-pulse solutions to the stochastic fitzhugh–nagumo equations

Eichinger, Katharina (Université Paris-Dauphine)
Gnann, M.V. (TU Delft Analysis)
Kuehn, Christian (Technische Universität München)

2022

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 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 Krüger and Stannat (Nonlinear Anal. 162 (2017) 197–223) 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.

FitzHugh–Nagumo equations
pulse
stability
stochastic reaction–diffusion equations
traveling waves

http://resolver.tudelft.nl/uuid:544ac73d-9b05-4593-a038-f21a3687ad0d

https://doi.org/10.1214/21-AAP1759

2023-07-01

1050-5164

Annals of Applied Probability, 32 (5), 3229-3282

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

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© 2022 Katharina Eichinger, M.V. Gnann, Christian Kuehn