Evolution systems of measures for stochastic flows

A new concept of an evolution system of measures for stochastic flows is considered. It corresponds to the notion of an invariant measure for random dynamical systems (or cocycles). The existence of evolution systems of measures for asymptotically compact stochastic flows is obtained. For a white noise stochastic flow, there exists a one to one correspondence between evolution systems of measures for a stochastic flow and evolution systems of measures for the associated Markov transition semigroup. As an application, an alternative approach for evolution systems of measures of 2D stochastic Navier–Stokes equations with a time-periodic forcing term is presented.     

Hitting times for random dynamical systems

The purpose of this article is to study the hitting times for random dynamical systems. For general systems we give a lower bound in terms of the local dimension. For fast mixing systems we obtain an equality. Moreover, under a power law decay of correlations we obtain lower and upper bounds of the hitting times for absolutely continuous stationary measures.     

Random chain recurrent sets for random dynamical systems

It is known by the Conley's theorem that the chain recurrent set CR(?) of a deterministic flow ? on a compact metric space is the complement of the union of sets B(A) ? A, where A varies over the collection of attractors and B(A) is the basin of attraction of A. It has recently been shown that a similar decomposition result holds for random dynamical systems (RDSs) on non-compact separable complete metric spaces, but under a so-called absorbing condition. In the present article, the authors introduce a notion of random chain recurrent sets for RDSs, and then prove the random Conley's theorem on non-compact separable complete metric spaces without the absorbing condition.     

Centre manifolds for infinite dimensional random dynamical systems

Stochastic centre manifolds theory are crucial in modelling the dynamical behaviour of complex systems under stochastic influences. The existence of stochastic centre manifolds for infinite dimensional random dynamical systems is shown under the assumption of exponential trichotomy. The theory provides a support for the discretisations of nonlinear stochastic partial differential equations with space–time white noise.