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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 49 (12 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
Stochastic dissipative PDE's and Gibbs measures
 Comm. Math. Phys
, 2000
"... We study a class of dissipative nonlinear PDE's forced by a random force # # (t, x), with the space variable x varying in a bounded domain. The class contains the 2D NavierStokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistic ..."
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Cited by 39 (10 self)
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We study a class of dissipative nonlinear PDE's forced by a random force # # (t, x), with the space variable x varying in a bounded domain. The class contains the 2D NavierStokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in x and stationary, shortcorrelated in time t. In this paper, we confine ourselves to "kick forces" of the form # # (t, x) = +# X k=# #(t  kT )#k (x), where the #k 's are smooth bounded identically distributed random fields. The equation in question defines a Markov chain in an appropriately chosen phase space (a subset of a function space) that contains the zero function and is invariant for the (random) flow of the equation. Concerning this Markov chain, we prove the following main result (see Theorem 2.2): The Markov chain has a unique invariant measure. To prove this theorem, we present a construction assigning, to any invariant...
Gibbsian Dynamics and Ergodicity for the Stochastically Forced NavierStokes Equation
, 2000
"... We study stationary measures for the twodimensional NavierStokes equation with periodic boundary condition and random forcing. We prove uniqueness of the stationary measure under the condition that all “determining modes” are forced. The main idea behind the proof is to study the Gibbsian dynamics ..."
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Cited by 38 (13 self)
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We study stationary measures for the twodimensional NavierStokes equation with periodic boundary condition and random forcing. We prove uniqueness of the stationary measure under the condition that all “determining modes” are forced. The main idea behind the proof is to study the Gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the timehistory of the low modes.
Exponential Convergence for the Stochastically Forced NavierStokes Equations and Other Partially Dissipative Dynamics
, 2002
"... We prove that the two dimensional NavierStokes equations possesses an exponentially attracting invariant measure. This result is in fact the consequence of a more general "Harrislike" ergodic theorem applicable to many dissipative stochastic PDEs and stochastic processes with memory. A simple iter ..."
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Cited by 34 (9 self)
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We prove that the two dimensional NavierStokes equations possesses an exponentially attracting invariant measure. This result is in fact the consequence of a more general "Harrislike" ergodic theorem applicable to many dissipative stochastic PDEs and stochastic processes with memory. A simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting. To analyze the iterated map, a general "Doeblinlike" theorem is proven. One of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the nonMarkovian processes.
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 15 (7 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
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 11 (6 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:
The Dissipative Scale Of The Stochastics Navier Stokes Equation: Regularization And Analyticity
 J. STATIST. PHYS
, 2002
"... We prove spatial analyticity for solutions of the stochastically forced Navier Stokes equation, provided that the forcing is sufficiently smooth spatially. We also give estimates, which extend to the stationary regime, providing strong control of both of the expected rate of dissipation and fluctuat ..."
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Cited by 7 (4 self)
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We prove spatial analyticity for solutions of the stochastically forced Navier Stokes equation, provided that the forcing is sufficiently smooth spatially. We also give estimates, which extend to the stationary regime, providing strong control of both of the expected rate of dissipation and fluctuations about this mean. Surprisingly, we could not obtain nonrandom estimates of the exponential decay rate of the spatial Fourier spectra.
Pathwise stationary solutions of stochastic Burgers equations with L²[0, 1]noise and stochastic Burgers integral equations on infinite horizon
, 2006
"... ..."
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 4 (4 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.
Stationary Solutions of Stochastic Differential Equation with Memory and Stochastic Partial Differential Equations
, 2003
"... 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 soluti ..."
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Cited by 4 (1 self)
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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 NavierStokes equation and stochastic GinsburgLandau equation.