Analysis of the Stochastic Fitzhugh–nagumo System

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 ν associated with the transition semigroup Pt; further, we identify its infinitesimal generator in the space L 2 (H; ν).

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

“…However, there are additional new phenomena possible if we consider systems for d ≥ 2 such as spiral-like structures [167,44]. It is natural to conjecture that spiral-like waves can be found if we perturb the classical models for spiral waves such as the FitzHugh-Nagumo [57,134] equation, the Barkley model [11], or the Oregonator system by noise [15,17,110,160,167].…”

Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition

“…Then, a confidence interval for the estimationp n Q,κ−ε of p Q,κ with confidence level α is given by [p n Q,κ−ε − γ,p n Q,κ−ε + γ] where γ = (αm) − 1 2 1 + 4ε −2 C 3.1 n −1 Remark. The additional regularity of the solution may be obtained in the context of mild solutions as in [1] if Q is sufficiently regular, since the heat semigroup in the equation for v is analytic and therefore maps H to D(A) = H 2 (0, 1). However, this needs further investigation.…”

Simulation of SPDEs for Excitable Media Using Finite Elements

“…We assume that W Q is a Q-Wiener process with an operator Q which satisfies Assumption 1. We impose the following condition on the operator A in (8).…”

“…Noisy FHN model and especially, FHN with white noise, have been extensively studied. We refer the reader to [8] where all the arguments needed to prove the following proposition are developed. The proof of this proposition relies on Itô Formula, see Chapter 1, Section 4.5 of [15], and the fact that the functional defined by…”