The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L 2 0(T 2). Unlike in previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds under a Hörmander-type condition. This requires some interesting non-adapted stochastic analysis. 1

We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions amount essentially to the fact that the equation transmits the noise to all its determining modes. Several examples are investigated, including some where the noise does not act on every determining mode directly.

We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations. Since our desire is to make the core ideas clear, we will mostly work with a specific example: the stochastically forced Navier Stokes equations. To further clarify ideas, we will also examine in detail a toy problem. A few general theorems are given. Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and hypoellipticity are all discussed. This article attempts to collect a number of ideas which have proven useful in the study of stochastically forced dissipative partial differential equations. The discussion will center around those of ergodicity but will also touch on the regularity of both solutions and transition densities. Since our desire is to make the core ideas clear, we will mostly work with a specific example: the stochastically forced Navier Stokes equations. To further clarify ideas, we will also examine in detail a toy problem. Though we have not tried to give any great generality, we also present a number of abstract results to help isolate what assumptions are used in which arguments. Though a few results are presented in new ways and a number of proofs are streamlined, the core ideas remain more or less the same as in the originally cited papers. We do improve sightly the exponential mixing results given in [Mat02c]; however, the techniques used are the same. Lastly, we do not claim to be exhaustive. This is not meant to be an all encompassing review article. The view point given here is a personal one; nonetheless, citations are given to good starting points for related works both by the author and others. Consider the two-dimensional Navier-Stokes equation with stochastic forcing:

We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł p-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the two-dimensional stochastic Navier-Stokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic Navier-Stokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1

We consider the 2D Navier–Stokes system, perturbed by a random force, such that sufficiently many of its Fourier modes are excited (e.g. all of them are). We discuss the results on the existence and uniqueness of a stationary measure for this system, obtained in last years, homogeneity of the measures and some their limiting properties. Next we use these results to prove that solutions of the equations obey the central limit theorem and the strong law of large numbers. Keywords: 1.

Dedicated to M. I. Vishik on the occasion of his 80th birthday. Abstract. For a class of random dynamical systems which describe dissipative nonlinear PDEs perturbed by a bounded random kick-force, we propose a “direct proof ” of the uniqueness of the stationary measure and exponential convergence of solutions to this measure, by showing that the transfer operator acting in the space of probability measures given the Kantorovich metric, defines a contraction of this space. Next we use results of [Kuk97, Kuk99] to study properties of this measure in the turbulence limit (as the viscosity goes to zero), for some nonlinear PDEs. In [KS00] 1 (see also [KS02]) A. Shirikyan and the author of this paper considered a class of nonlinear dissipative PDEs perturbed by smooth in space random forces. We proved that these equations, treated as random dynamical systems in a function space, have unique stationary measures. The forces considered in [KS00]

We explore Ito stochastic differential equations where the drift term has possibly infinite dependence on the past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proved if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation.

The paper deals with infinite-dimensional random dynamical systems. Under the condition that the system in question is of mixing type and possesses a random compact attracting set, we show that the support of the unique invariant measure is the minimal random point attractor. The results obtained apply to the randomly forced 2D Navier–Stokes system.

The paper deals with infinite-dimensional random dynamical systems. Under the

Abstract. It is shown that, on certain weighted spaces of vector fields on R 3, any homogeneous measure of finite energy density and dissipation can be approximated in the second Wasserstein distance by homogeneous measures supported by finite trigonometric polynomials of increasing period and degree. In particular, the periodic correlation functions of the approximation converge uniformly on compact sets of R 3 to the correlation function of the given measure. 1 2 S. DOSTOGLOU AND J.D. KAHL 1.