Random iterations of maps on ${\mathbb{R}}^k$: asymptotic stability, synchronization and functional central limit theorem

Abstract

We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset $S$ of the Euclidean space ${\mathbb{R}}^k$. We assume the maps are monotone (with respect to a suitable partial order) and a "topological" condition on the maps. Then, we prove the existence of a pullback random attractor whose distribution is the unique stationary measure of the random iteration, and we obtain the synchronization of random orbits. As a consequence of the synchronization phenomenon, a functional central limit theorem is established.