Lattice Approximation for Stochastic Reaction Diffusion Equations with One-Sided Lipschitz Condition

TL;DR

This paper studies the strong convergence of finite difference spatial approximations for stochastic reaction-diffusion equations with multiplicative noise under a one-sided Lipschitz condition, providing explicit convergence rates based on solution regularity.

Contribution

It introduces a convergence analysis for finite difference methods under a one-sided Lipschitz condition, with explicit rates depending on solution regularity, and applies it to noisy FitzHugh-Nagumo systems.

Findings

Convergence rate depends on the regularity of the exact solution.

Explicit convergence rates are derived for solutions with higher regularity.

Application to stochastic FitzHugh-Nagumo systems demonstrates practical relevance.

Abstract

We consider strong convergence of the finite differences approximation in space for stochastic reaction diffusion equations with multiplicative noise under a one-sided Lipschitz condition only. We derive convergence with an implicit rate depending on the regularity of the exact solution. This can be made explicit if the variational solution has more than its canonical spatial regularity. As an application, spatially extended FitzHugh-Nagumo systems with noise are considered.