Travelling waves for discrete stochastic bistable equations

TL;DR

This paper investigates how nonlocal discretization and stochastic noise affect travelling wave solutions in a discrete Nagumo equation, showing that solutions remain close to classical fronts under certain conditions.

Contribution

It demonstrates the existence and stability of travelling wave solutions in discrete stochastic Nagumo equations, extending understanding of their behavior under noise and discretization.

Findings

Solutions stay close to classical Nagumo fronts with high probability

Traveling waves exist under small noise and discretization levels

Results apply to a broad class of discrete stochastic bistable systems

Abstract

Many physical, chemical and biological systems have an inherent discrete spatial structure that strongly influences their dynamical behaviour. Similar remarks apply to internal or external noise, as well as to nonlocal coupling. In this paper we study the combined effect of nonlocal spatial discretization and stochastic perturbations on travelling waves in the Nagumo equation, which is a prototypical model for bistable reaction-diffusion partial differential equations (PDEs). We prove that under suitable parameter conditions, various discrete-stochastic variants of the Nagumo equation have solutions, which stay close on long time scales to the classical monotone Nagumo front with high probability if the noise level and spatial discretization are sufficiently small.