Remarks on Ergodic Theory of Stochastic Flows and Control Flows

Abstract

Stochastic systems and control systems with values in a state space M can be considered as dynamical systems on the space U × M, where U denotes the space of admissible control functions for control systems, and the trajectory space of an underlying noise process for stochastic systems. Invariant probability measures for these flows are the main topic of this paper: We show that their support is contained in sets D ⊂ U × M, which are the lifts of so-called control sets D ⊂ M to invariant sets in U × M. Several results on the characterization of control sets D are given, together with criteria for the existence of invariant measures u on U × M with supp μ ⊂ V. The case of Markovian stochastic systems is treated in some detail. Because of the importance in applications, we prove rather complete results for two classes of systems: linearized systems, which play a crucial role in the theory of Lyapunov exponents for stochastic and control flows, and general nonlinear systems with one dimensional state space, which are important in stochastic bifurcation theory.