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Ergodicity of the 2D NavierStokes equations with degenerate forcing, preprint
"... The stochastic 2D NavierStokes 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 ..."
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Cited by 116 (18 self)
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The stochastic 2D NavierStokes 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 Hörmandertype condition. This requires some interesting nonadapted stochastic analysis. 1
Exponential Mixing Properties of Stochastic PDEs Through Asymptotic Coupling
 Probab. Theory Related Fields
, 2001
"... 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 ..."
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Cited by 48 (8 self)
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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.
Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... 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 ptype norm, but involves the derivative of the observable as ..."
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Cited by 40 (15 self)
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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 ptype 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 infinitedimensional 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 twodimensional stochastic NavierStokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic NavierStokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1
A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs
"... We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this ..."
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Cited by 32 (14 self)
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We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator Mt can be obtained. Informally, this bound can be read as “Fix any finitedimensional projection Π on a subspace of sufficiently regular functions. Then the eigenfunctions of Mt with small eigenvalues have only a very small component in the image of Π.” We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of “Wiener polynomials, ” where the coefficients are possibly nonadapted stochastic processes satisfying a Lipschitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris ’ lemma, which is unavailable in the present context. We conclude by showing that the twodimensional stochastic NavierStokes equations and a large class of reactiondiffusion equations fit the framework of our theory. Contents 1
On recent progress for the stochastic Navier Stokes equations
 In Journées Équations aux dérivées partielles, ForgeslesEaux, XI:1–52, 2003. see http://www.math.sciences.univnantes.fr/edpa/2003/html/. [MY02] [Pro90] [Sin94] Nader Masmoudi and LaiSang
"... 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 examp ..."
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Cited by 22 (7 self)
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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 twodimensional NavierStokes equation with stochastic forcing:
Exponential mixing for 2D NavierStokes equations perturbed by an unbounded noise
 J. Math. Fluid Mech
"... The paper is devoted to the problem of mixing for twodimensional Navier–Stokes equations perturbed by an unbounded kick force. We develop the coupling approach suggested in [16] to show that any solution exponentially converges to the stationary measure in the dual Lipschitz norm. This property co ..."
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Cited by 21 (6 self)
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The paper is devoted to the problem of mixing for twodimensional Navier–Stokes equations perturbed by an unbounded kick force. We develop the coupling approach suggested in [16] to show that any solution exponentially converges to the stationary measure in the dual Lipschitz norm. This property complements some earlier results established in [15] for the same model.
Memory loss for timedependent dynamical systems
 Math. Res. Lett
"... Abstract. This paper discusses the evolution of probability distributions for certain timedependent dynamical systems. Exponential loss of memory is proved for expanding maps and for onedimensional piecewise expanding maps with slowly varying parameters. 1. ..."
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Cited by 16 (8 self)
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Abstract. This paper discusses the evolution of probability distributions for certain timedependent dynamical systems. Exponential loss of memory is proved for expanding maps and for onedimensional piecewise expanding maps with slowly varying parameters. 1.
Ergodic theorems for 2D statistical hydrodynamics
 Rev. Math. Phys
, 2002
"... 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 measur ..."
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Cited by 12 (1 self)
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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.
Ergodicity for a class of Markov processes and applications to randomly forced PDE’s II
 Discrete Contin. Dyn. Syst. Ser. B
, 2006
"... Abstract. The paper is devoted to studying the problem of ergodicity for dissipative PDE’s perturbed by an external random force. We give a simple sufficient condition for uniqueness and stability of a stationary measure for a family of Markov processes. The result is applied to the 2D Navier–Stokes ..."
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Cited by 11 (2 self)
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Abstract. The paper is devoted to studying the problem of ergodicity for dissipative PDE’s perturbed by an external random force. We give a simple sufficient condition for uniqueness and stability of a stationary measure for a family of Markov processes. The result is applied to the 2D Navier–Stokes system. The problem of ergodicity for randomly forced PDE’s of mathematical physics was in the focus of attention during the last ten years. The first result in this direction was obtained by Flandoli and Maslowski [FM95], who studied the 2D Navier–Stokes (NS) system perturbed by a random force that is white in time and sufficiently irregular in the space variables x. They proved that the
Nonstationary compositions of Anosov diffeomorphisms
 Nonlinearity
"... Abstract. Motivated by nonequilibrium phenomena in nature, we study dynamical systems whose timeevolution is determined by nonstationary compositions of chaotic maps. The constituent maps are topologically transitive Anosov diffeomorphisms on a 2dimensional compact Riemannian manifold, which are ..."
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Cited by 10 (6 self)
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Abstract. Motivated by nonequilibrium phenomena in nature, we study dynamical systems whose timeevolution is determined by nonstationary compositions of chaotic maps. The constituent maps are topologically transitive Anosov diffeomorphisms on a 2dimensional compact Riemannian manifold, which are allowed to change with time — slowly, but in a rather arbitrary fashion. In particular, such systems admit no invariant measure. By constructing a coupling, we prove that any two sufficiently regular distributions of the initial state converge exponentially with time. Thus, a system of the kind loses memory of its statistical history rapidly.