  We investigate the stability of traveling-pulse solutions to the stochastic
FitzHughNagumo 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 Kr\"uger 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 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.
