Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions

TL;DR

This paper investigates the mean-square stability of numerical approximations for infinite-dimensional stochastic differential equations, providing conditions for stability and illustrating results with simulations of the stochastic heat equation.

Contribution

It offers necessary and sufficient stability conditions for various discretization schemes applied to infinite-dimensional SDEs, linking numerical stability to analytical solution properties.

Findings

Stability conditions are established for spectral Galerkin, finite element, and common time-stepping schemes.

Theoretical results are validated through simulations of the stochastic heat equation.

A connection between numerical and analytical stability properties is demonstrated.

Abstract

The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory.