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37
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
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.
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 25 (7 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 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:
Ergodicity of the finite dimensional approximation of the 3d navier–stokes equations forced by a degenerate
, 2002
"... Abstract. We prove ergodicity of the finite dimensional approximations of the three dimensional NavierStokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to ..."
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Cited by 11 (1 self)
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Abstract. We prove ergodicity of the finite dimensional approximations of the three dimensional NavierStokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential. 1.
Ergodicity of Stochastic Differential Equations Driven by Fractional Brownian Motion
, 2003
"... We study the ergodic properties of finitedimensional systems of SDEs driven by nondegenerate additive fractional Brownian motion with arbitrary Hurst parameter H ∈ (0, 1). A general framework is constructed to make precise the notions of “invariant measure ” and “stationary state ” for such a syste ..."
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Cited by 11 (2 self)
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We study the ergodic properties of finitedimensional systems of SDEs driven by nondegenerate additive fractional Brownian motion with arbitrary Hurst parameter H ∈ (0, 1). A general framework is constructed to make precise the notions of “invariant measure ” and “stationary state ” for such a system. We then prove under rather weak dissipativity conditions that such an SDE possesses a unique stationary solution and that the convergence rate of an arbitrary solution towards the stationary one is (at least) algebraic. A lower bound on the exponent is also given.
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 8 (7 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
Metastability in Interacting Nonlinear Stochastic Differential Equations II: LargeN Behaviour
, 2006
"... We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong co ..."
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Cited by 6 (3 self)
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We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order N 2), the system synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. In a previous paper, we showed that the transition from strong to weak coupling involves a sequence of symmetrybreaking bifurcations of the system’s stationary configurations, and analysed in particular the behaviour for coupling intensities slightly below the synchronisation threshold, for arbitrary N. Here we describe the behaviour for any positive coupling intensity γ of order N 2, provided the particle number N is sufficiently large (as a function of γ/N 2). In particular, we determine the transition time between synchronised states, as well as the shape of the “critical droplet ” to leading order in 1/N. Our techniques involve the control of the exact number of periodic orbits of a nearintegrable twist map, allowing us to give a detailed description of the system’s potential landscape, in which the metastable behaviour is encoded.
On exponential convergence to a stationary measure for nonlinear PDEs, perturbed by random kickforces
 and the turbulencelimit, The M. I. Vishik Moscow PDE Seminar
, 2002
"... 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 kickforce, we propose a “direct proof ” of the uniqueness of the stationary measure and exponential convergence o ..."
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Cited by 5 (1 self)
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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 kickforce, 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]