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

Abstract: We study a damped stochastic non-linear Schrödinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markovian transition semi-group toward a unique invariant probability measure. This kind of method was originally developped to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schrödinger equation in the one dimensional cubic case. We prove that the mixing property holds and that the rate of convergence to equilibrium is at least polynomial of any power. Key words: Non-linear Schrödinger equations, Markovian transition semi-group, invariant measure, ergodicity, coupling method, Girsanov’s formula, expectational

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.

Abstract: We establish a general criterion which ensures exponential mixing of parabolic Stochastic Partial Differential Equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two representative examples: 2D Navier-Stokes (NS) equations and Complex Ginzburg-Landau (CGL) equation with a locally Lipschitz noise. Due to the possible degeneracy of the noise, Doob theorem cannot be applied. Hence a coupling method is used in the spirit of [11], [23] and [25]. Previous results require assumptions on the covariance of the noise which might seem restrictive and artificial. For instance, for NS and CGL, the covariance operator is supposed to be diagonal in the eigenbasis of the Laplacian and not depending on the high modes of the solutions. The method developped in the present paper gets rid of such assumptions and only requires that the range of the covariance operator contains the low modes.

Abstract: We study the Navier-Stokes equations in dimension 3 (NS3D) driven by a noise which is white in time. We establish that if the noise is at same time sufficiently smooth and non degenerate in space, then the weak solutions converge exponentially fast to equilibrium. We use a coupling method. The arguments used in dimension two do not apply since, as is well known, uniqueness is an open problem for NS3D. New ideas are introduced. Note however that many simplifications appears since we work with non degenerate noises. Key words: Stochastic three-dimensional Navier-Stokes equations, Markov transition semi-group, invariant measure, ergodicity, coupling method, exponential mixing,

We consider random perturbations of 2D Navier--Stokes equations.

Abstract. We consider a stochastic Klein-Gordon wave equation modeling heat flow in a linear field that is coupled to thermal reservoirs at different temperatures. We discuss, in a perturbative context, the approach to a stationary, non-equilibrium state, and show that the state is supported on field configurations which are Hölder continuous, with any exponent less than 1/2. We determine the heat flux to lowest order in perturbation theory. 1.

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

Abstract. J. Math. Fluid Mechanics, 2004, in press. The authors consider stochastic aspects of the stabilization problem for two and three-dimensional Oseen equations with help of feedback control defined on a part of the fluid boundary. Stochastic issues arise when inevitable unpredictable fluctuations in numerical realization of stabilization procedures are taken into account and they are supposed to be independent identically distributed random variables. Under this assumption the solution to the stabilization problem obtained via boundary feedback control can be described by a Markov chain or a discrete random dynamical system. It is shown that this random dynamical system possesses a unique, exponentially attracting, invariant measure, namely, this random dynamical system is ergodic. This gives adequate statistical description of the stabilization process on the stage when stabilized solution has to be retained near zero (i.e. near unstable state of equilibrium).

Recently, a number of authors have investigated the conditions under which a stochastic perturbation acting on an infinite dimensional dynamical system, e.g. a partial differential equation, makes the system ergodic and mixing. In particular, one is interested in finding minimal and physically natural conditions on the nature of the stochastic perturbation. I shall review recent results on this question; in particular, I shall discuss the Navier-Stokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity ν, and grows like ν −3 when ν goes to zero. This Markov process has a unique invariant measure and is exponentially mixing in time. 2000 Mathematics Subject Classification: 35Q30, 60H15. Keywords and Phrases: Navier-Stokes equations with random perturbations, Markov approximations, Statistical mechanics of one-dimensional systems. 1.