Flows of stochastic dynamical systems: ergodic theory

Abstract

We present a version of the Multiplicative Ergodic (Oseledec) Theorem for the flow of a nonlinear stochastic system definedon a smooth compact manifold. This theorem establishes the existence of a Lyapunov spectrum for the flow, which characterises the asymptotic behaviour of the derivative flow. Then we establish the existence of stable manifolds for the flow (on which trajectories cluster) associated with the Lyapunov spectrum. This work is a generalisation of that of Ruelle who deals with ordinary dynamical systems. Finally we give an example of a stochastic system for which the flow is calculated explicitly, and which illustrates the behaviour predicted by the abstract results.