PATHWISE GLOBAL ATTRACTORS FOR STATIONARY RANDOM DYNAMIC-SYSTEMS

The asymptotic behaviour of random dynamical systems in Polish spaces is considered. Under the assumption of existence of a random compact absorbing set, assumption supposed to hold path by path, a candidate pathwise attractor A(omega) is defined. The goal of the paper is to show that, in the case of stationary dynamical systems, A(omega) attracts bounded sets, is measurable with respect to the sigma-algebra of invariant sets, and is independent of omega when the system is ergodic. An application to a general class of Navier-Stokes type equations perturbed by a multiplicative ergodic real noise is discussed in detail.