Approximating the invariant measures of randomly perturbed dissipative maps (Q1916798)

Let \(S:X\to X\) be a Lipschitz mapping of a manifold \(X\) or a subset of \(\mathbb{R}^d\). The author considers random dynamical systems \(x_{n+1}= Sx_n+\varepsilon\xi_n\), where \(0<\varepsilon\) is very small and \(\xi_n\) a sequence of independent identically distributed random variables. 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 \(S\) has strong contraction properties. It is also shown that the invariant measure, resp. the stationary measure of the associated Markov chain, are approached exponentially. The proofs rest on an early result of Yosida-Kakutani and on a spectral approximation due to Chatelin. There are examples of implementation on one- and two-dimensional maps.