Non-normal drift structures and linear stability analysis of numerical methods for systems of stochastic differential equations

We investigate mean-square asymptotic stability of equilibria in linear systems of stochastic differential equations with non-normal drift coefficients, with particular emphasis on the role of interactions between the drift and diffusion structures that act along, orthogonally to, and laterally to the flow. Hence we construct test systems with non-normal drift coefficients and characteristic diffusion structures for the purposes of a linear stability analysis of the θθ-Maruyama method. Next we discretise these test systems and examine the mean-square asymptotic stability of equilibria of the resulting systems of stochastic difference equations. Finally we indicate how this approach may help to shed light on numerical discretisations of stochastic partial differential equations with multiplicative space–time perturbations.