Invariant Measures and Some Ergodic Theory for Random Dynamical Systems

In the present chapter the results obtained before are going to be applied to random dynamical systems (RDS). We will give only the abstract characterisation of RDS. For a comprehensive and thorough description of the generation of RDS from random and stochastic differential equations in finite dimensional spaces we refer to Arnold [I], in particular to Chapter 2. Concerning the generation of RDS from infinite-dimensional problems, in particular from stochastic parabolic stochastic partial differential equations on bounded domains, ser Flandoli [23]. The results we are interested here are assertions about existence of invariant measures and of invariant lLIarkov measures for RDS on Polish spaces, as well as a cllaracterisation of the convergence of time means in terms of integrals over ergodic invariant measures. These results are the ones which need the topological prerequisites obtained in previous chapters.