Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions
For researchers in numerical analysis of stochastic PDEs, this work offers rigorous stability criteria for a broad class of methods, though it is an incremental extension of finite-dimensional stability theory.
The paper provides necessary and sufficient conditions for mean-square stability of numerical approximations to infinite-dimensional stochastic differential equations, applying them to common discretization schemes and illustrating with the stochastic heat equation.
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.