An adaptive subdivision technique for the approximation of attractors and invariant measures: Proof of convergence

We prove convergence of a recently introduced adaptive multilevel algorithm for the efficient computation of invariant measures and attractors of dynamical systems. The proof works in the context of (sufficiently regular) stochastic processes and essentially shows that the discretization of phase space leads to a small random perturbation of the original process. A generalized version of a lemma of Khasminskii then gives the desired result.