University of Heidelberg, Applied Analysis Seminar
Thu 07.05.2020
14:15, posted in Applied Analysis
Prof. Dr. Manuel Gnann (Technische Universiteit Delft)
Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations
Online via HeiCONF

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 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.

The talk is based on joint work with Katharina Eichinger (CEREMADE,
Université Paris-Dauphine) and Christian Kuehn (Technische Universität

<< Back