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23
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 17 (8 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
ERGODICITY FOR THE WEAKLY DAMPED STOCHASTIC NONLINEAR SCHRÖDINGER EQUATIONS
, 2004
"... Abstract: We study a damped stochastic nonlinear 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 semigroup toward a unique invariant probability measure. This kind of meth ..."
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Cited by 11 (7 self)
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Abstract: We study a damped stochastic nonlinear 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 semigroup toward a unique invariant probability measure. This kind of method was originally developped to prove exponential mixing for strongly dissipative equations such as the NavierStokes 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: Nonlinear Schrödinger equations, Markovian transition semigroup, invariant measure, ergodicity, coupling method, Girsanov’s formula, expectational
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 9 (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.
Exponential Mixing for Stochastic PDEs: The NonAdditive Case, preprint available on http://www.bretagne.enscachan.fr/math/people/cyril.odasso
"... 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 NavierStokes (NS) equations ..."
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Cited by 6 (5 self)
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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 NavierStokes (NS) equations and Complex GinzburgLandau (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.
Exponential mixing for the 3D stochastic Navier–Stokes equations
, 2006
"... Abstract: We study the NavierStokes 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. ..."
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Cited by 5 (0 self)
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Abstract: We study the NavierStokes 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 threedimensional NavierStokes equations, Markov transition semigroup, invariant measure, ergodicity, coupling method, exponential mixing,
Ergodicity and Mixing for Stochastic Partial Differential Equations
, 2002
"... 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 natur ..."
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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 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 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: NavierStokes equations with random perturbations, Markov approximations, Statistical mechanics of onedimensional systems. 1.
Some limiting properties of randomly forced 2D NavierStokes equations
"... We consider random perturbations of 2D NavierStokes equations. ..."
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Cited by 3 (1 self)
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We consider random perturbations of 2D NavierStokes equations.
On Random Attractors for MixingType Systems
, 2002
"... The paper deals with infinitedimensional random dynamical systems. Under the ..."
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Cited by 1 (0 self)
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The paper deals with infinitedimensional random dynamical systems. Under the
unknown title
, 2003
"... Low regularity solutions to a gently stochastic nonlinear wave equation in nonequilibrium statistical mechanics ..."
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Low regularity solutions to a gently stochastic nonlinear wave equation in nonequilibrium statistical mechanics
ON ERGODICITY OF SOME MARKOV PROCESSES
, 810
"... To the memory of Andrzej Lasota Abstract. We formulate a criterion for the existence, uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, the weak ∗ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process st ..."
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To the memory of Andrzej Lasota Abstract. We formulate a criterion for the existence, uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, the weak ∗ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting with any initial distribution, is established. The principal assumptions are the lower bound of the ergodic averages of the transition probability function and the eproperty of the semigroup. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. We use the weak ∗ mean ergodicity of the respective invariant probability measure to derive the law of large numbers for the trajectory of the passive tracer. 1.