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48
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 48 (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
Uniqueness of the Invariant Measure for a Stochastic PDE Driven by Degenerate Noise
, 2001
"... We consider the stochastic GinzburgLandau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The lowlying frequencies are then only connected to this forcing through the nonlinear (cubic) term of the GinzburgLandau equation. Under these assumpti ..."
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Cited by 40 (11 self)
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We consider the stochastic GinzburgLandau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The lowlying frequencies are then only connected to this forcing through the nonlinear (cubic) term of the GinzburgLandau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for the lowlying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies the uniqueness of that measure. Contents 1 Introduction 2 2 Some Preliminaries on the Dynamics 5 3 Controllability 6 4 Strong Feller Property and Proof of Theorem 1.1 9 5 Regularity of the Cutoff Process 11 5.1 Splitting and Interpolation Spaces . . . . . . . . . . . . . . . . . . . 12 5.2 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.3 Smoothing Properties of the...
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 38 (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 37 (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 33 (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.
Ergodicity of the 2D NavierStokes Equations with Random Forcing
 Comm. Math. Phys
, 2000
"... We consider the NavierStokes equation on a two dimensional torus with a random force, acting at discrete times and analytic in space, for arbitrarily small viscosity coefficient. We prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time. 1 ..."
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Cited by 29 (2 self)
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We consider the NavierStokes equation on a two dimensional torus with a random force, acting at discrete times and analytic in space, for arbitrarily small viscosity coefficient. We prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time. 1 Introduction A convenient mathematical model for the study of homogenous isotropic turbulence is to consider the NavierStokes equation subject to a random stationary (in space and time) forcing. The turbulent situation is modelled by a smooth force, i.e. one whose Fourier transform decays fast for large wave numbers. One is then interested in various properties of the correlation functions of the velocity field in a stationary state of the ensuing stochastic process. An obvious first question concerns the large time convergence to such a stationary state starting from an arbitrary initial condition of the velocity field, i.e. the uniqueness of the stationary state. In this paper we prove t...
Exponential Mixing of the 2D Stochastic NavierStokes Dynamics
 Comm. Math. Phys
, 2000
"... We consider the NavierStokes 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 , and grows like \Gamma3 when goes to zero. We prove that this Markov process has a ..."
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Cited by 23 (1 self)
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We consider the NavierStokes 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 , and grows like \Gamma3 when goes to zero. We prove that this Markov process has a unique invariant measure and is exponentially mixing in time. 1 Introduction Homogenous isotropic turbulence is often mathematically modelled by Navier Stokes equation subjected to an external stochastic driving force which is stationary in space and time and "large scale", which in particular means smooth in space. The status of the existence and uniqueness of solutions to the stochastic PDE parallels that of the deterministic one. In particular, in two dimensions, it holds under very general conditions. However, for physical reasons, one is interested in the existence, uniqueness and properties of the stationary state of the resulting Markov process. While the existence of such a state fo...
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
Lefevere: Probabilistic estimates for the two dimensional stochastic NavierStokes equations
 J. Stat. Phys
"... We consider the NavierStokes equation on a two dimensional torus with a random force, white noise in time and analytic in space, for arbitrary Reynolds number R. We prove probabilistic estimates for the long time behaviour of the solutions that imply bounds for the dissipation scale and energy spec ..."
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Cited by 14 (3 self)
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We consider the NavierStokes equation on a two dimensional torus with a random force, white noise in time and analytic in space, for arbitrary Reynolds number R. We prove probabilistic estimates for the long time behaviour of the solutions that imply bounds for the dissipation scale and energy spectrum as R → ∞. 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: