Analysis of the stochastic FitzHugh-Nagumo system

TL;DR

This paper investigates the stochastic FitzHugh-Nagumo system, establishing fundamental properties like existence, uniqueness, and long-term behavior, including the existence of an invariant measure and its generator, relevant for neurobiological modeling.

Contribution

It provides a rigorous analysis of the stochastic FitzHugh-Nagumo system, including existence, uniqueness, and characterization of invariant measures, advancing understanding of its asymptotic dynamics.

Findings

Existence and uniqueness of solutions

Existence of an invariant ergodic measure

Identification of the generator in L^2 space

Abstract

In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall be mainly concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure $\nu$ associated with the transition semigroup $P_t$; further, we identify its infinitesimal generator in the space $L^2(H;\nu)$.