Evolution systems of measures for stochastic flows

TL;DR

This paper introduces the concept of evolution systems of measures for stochastic flows, establishing their existence under certain conditions and exploring their relationship with Markov semigroups, with applications to stochastic Navier-Stokes equations.

Contribution

It defines a new concept for stochastic flows, proves existence results, and links these measures to Markov semigroups, providing an alternative approach for stochastic Navier-Stokes equations.

Findings

Existence of evolution systems of measures for asymptotically compact stochastic flows.

One-to-one correspondence between measures for white noise flows and Markov semigroups.

Application to 2D stochastic Navier-Stokes equations with periodic forcing.

Abstract

A new concept of {\em 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 \emph{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.