Random iterations of maps on Rk: asymptotic stability, synchronisation and functional central limit theorem

We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset S of the Euclidean space Rk<i. 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 synchronisation of random orbits. As a consequence of the synchronisation phenomenon, a functional central limit theorem is established.