Stability of random attractors under perturbation and approximation

The comparison of the long-time behaviour of dynamical systems and their numerical approximations is not straightforward since in general such methods only converge on bounded time intervals. However, one can still compare their asymptotic behaviour using the global attractor, and this is now standard in the deterministic autonomous case. For random dynamical systems there is an additional problem, since the convergence of numerical methods for such systems is usually given only on average. In this paper the deterministic approach is extended to cover stochastic differential equations, giving necessary and sufficient conditions for the random attractor arising from a random dynamical system to be upper semi-continuous with respect to a given family of perturbations or approximations.