We consider the stochastic FitzHugh-Nagumo equations, whose deterministic equivalent allows for fast and stable traveling-pulse solutions. In this talk, we investigate the stability of fast pulses in case of additive noise and derive a multiscale decomposition for small levels of the stochastic forcing. Our method is based on adapting the wave velocity by solving a stochastic ordinary differential equation and tracking perturbations of the wave meeting a stochastic partial differential equation coupled to an ordinary differential equation. Previous works have focused on applying this method to scalar equations, such as the stochastic Nagumo equation, which carry a self-adjoint structure. This structure is lost in case of the FitzHugh-Nagumo system and the linearization does not generate an analytic semigroup. We show that this problem can be overcome by making use of Riesz spectral projections in a certain way. This provides a major generalization as our approach appears to be applicable also to general stochastic nerve-axon equations, the stochastic periodically-forced NLS equation, or systems of stochastic reaction-diffusion equations.
The talk is based on joint work with Katharina Eichinger (CEREMADE, Université Paris Dauphine) and Christian Kuehn (Technical University of Munich).