Method of coupling in the theory of randomly forced Navier-Stokes equations

We describe a general approach enabling one to prove the uniqueness of a stationary distribution and a mixing property for the 2D Navier-Stokes system perturbed by a non-degenerate random force. It is based on a development of the classical coupling method introduced by Doeblin in 1940. We begin with the case of Markov chains in a phase space containing finitely many points. We next turn to the Navier-Stokes system and apply the Bogolyubov-Krylov argument to construct a stationary distribution. To prove the uniqueness and mixing, we introduce the concept of maximal coupling and use it to construct an auxiliary Markov process in the direct product of two copies of the original phase space. This auxiliary process is such that its marginal laws coincide with those of solutions for Navier-Stokes equations, and its components converge to each other as time goes to infinity. These properties imply the required results.