Approximating the Invariant Measures of Randomly Perturbed Dissipative Maps

We present a method for approximating the invariant measure of a randomly perturbed mapping S of Rd. Cases where the unperturbed mapping is singular are considered, as are cases where the sizes of random jumps are unbounded. Existence of an invariant measure and convergence of the scheme are proved when the mapping has strong contraction properties. The implementation is designed to be useful in experimental situations where the map is known only approximately and the distribution of the noise is unknown. Under the same contraction conditions, we also prove that the invariant measure, the stationary measure of the associated Markov chain, is approached exponentially under iteration of the perturbed map, independent of the starting point.