MULTISCALE ANALYSIS FOR TRAVELING-PULSE SOLUTIONS TO THE STOCHASTIC FITZHUGH-NAGUMO EQUATIONS

• 成果类型: Article
• 署名作者: Eichinger, Katharina; Gnann, Manuel V.; Kuehn, Christian
• 署名单位: Universite PSL; Universite Paris-Dauphine; Delft University of Technology; Technical University of Munich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1759
• 发表日期: 2022
• 页码: 3229-3282
• 关键词: nerve axon equations; reaction-diffusion equations; partial-differential-equations; tracking invariant-manifolds; homoclinic orbits; oscillatory tails; front propagation; random attractors; STABILITY; waves

摘要

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 Kruger 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.