Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems

Wu, Liming

doi:10.1016/s0304-4149(00)00061-2

Cited by 172 publications

(263 citation statements)

References 17 publications

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“…Lyapunov conditions are widely used to study the speed of convergence of Markov chains [33] or Markov processes [18], large or moderate deviations and essential spectral radii [45,47]. More recently, they have been used to study functional inequalities as weak Poincaré inequality [2] or super-Poincaré inequality [10].…”

Section: Lyapunov Function Conditionsmentioning

confidence: 99%

Transportation-information inequalities for Markov processes

Guillin

Léonard

et al. 2008

Probab. Theory Relat. Fields

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153

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Abstract. In this paper, one investigates the following type of transportation-information T c I inequalities: α(T c (ν, µ)) ≤ I(ν|µ) for all probability measures ν on some metric space (X , d), where µ is a given probability measure, T c (ν, µ) is the transportation cost from ν to µ with respect to some cost function c(x, y) on 2 . It is proved that W 2 I is stronger than Poincaré inequality, weaker than log-Sobolev inequality, and equivalent to it when Bakry-Emery's curvature is bounded from below. For the trivial metric cost d, one establishes the sharp transportation-information inequality W 1

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Section: Lyapunov Function Conditionsmentioning

confidence: 99%

Transportation-information inequalities for Markov processes

Guillin

Léonard

et al. 2008

Probab. Theory Relat. Fields

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153

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“…So the external force f is irregular with respect to the space variable, which is not very natural from the physical point of view. The proof is based on a general sufficient condition established in [23], and it essentially uses the strong Feller property. The main novelty of Theorem 1.1 is that it proves LDP without a lower bound on the coefficients b j as in (1.1) (so we do not have a strong Feller property).…”

Section: Xvii-4mentioning

confidence: 99%

Large deviations results for the stochastic Navier–Stokes equations

Nersesyan¹

2017

Séminaire Laurent Schwartz — EDP Et Applications

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In this note, we discuss the problem of large deviations for the stochastic 2D Navier-Stokes equations. We show that the occupation measures of the trajectories of the system satisfy a large deviations principle, provided that the noise acts on all Fourier modes. In the case when the noise is more degenerate and acts on all the determining modes, we obtain an LDP of local type. The proofs use the methods introduced in [13,20] based on a Kifer-type sufficient condition for LDP and a multiplicative ergodic theorem. AMS subject classifications: 35Q30, 60B12, 60F10Keywords: Navier-Stokes system, white-in-time noise, large deviations principle, occupation measures, coupling IntroductionIn this note, we review the results of the paper [21], where the large deviations principle (LDP) is studied for the stochastic 2D Navier-Stokes (NS) equations driven by a white-in-time noiseThis system describes the motion of an incompressible fluid in a bounded domain D ⊂ R 2 with a smooth boundary ∂D, where ν > 0 is the kinematic viscosity, u = (u 1 (t, x), u 2 (t, x)) and p = p(t, x) are unknown velocity field and pressure of the fluid, f is the external (random) force, and u,Before stating our results, let us make some comments about this system and recall some previous results on its ergodicity. The system is considered in the usual space of divergence-free vector fields where L = −Π∆ is the Stokes operator, B(u) = Π( u, ∇ u), and Π is the orthogonal projection onto H in L 2 (Leray projector).Structure of the noise. We assume that f is of the formwhere the functions h and η are, respectively, the deterministic and random components of the external force with h ∈ H andHere {b j } is a sequence in R + such that B 0 = ∞ j=1 b 2 j < ∞, {β j } is a sequence of independent standard Brownian motions defined on a filtered probability space (Ω, F, {F t }, P) satisfying the usual conditions (see Definition 2.25 in [14]), and {e j } is an orthonormal basis in H consisting of the eigenfunctions of L with eigenvalues {α j }.Under these conditions, problem (0.3), (0.2) admits a unique solution and defines a Markov family (u t , P u ) parametrised by the initial condition u = u 0 ∈ H. The corresponding Markov semigroups are given bywhere P t (u, Γ) = P u {u t ∈ Γ} is the transition function. Recall that a measure µ ∈ P(H) is stationary if P * t µ = µ for any t > 0. The existence of a stationary measure is a relatively simple question, it is proved by using the classical Bogolyubov-Krylov argument. The uniqueness and the mixing are much more difficult problems which have been extensively studied in recent years. We refer the reader to the papers [7,16,6,17,2,11,22] and the book [18] for a detailed discussion of this topic. In particular, as it is stated in the following theorem, if η is sufficiently non-degenerate, then the family (u t , P u ) admits a unique stationary measure, which is exponentially mixing (see Theorem 3.5.2 in [18]).1 To simplify the notation, we shall assume that ν = 1. 2 P(H) is the set of probability Borel measures...

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“…In [3] we obtain a slightly weaker result (the exponential decay for ||| . ||| q,2 for all q > 2) by using an ad-hoc Foster-Lyapunov function (see also the seminal paper [50]). …”

Section: The Meaning Of This Inequality Is That W Belongs To the Extementioning

confidence: 99%

Long time behavior of Markov processes

Cattiaux

2014

ESAIM: Proc.

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Abstract. These notes correspond to a three hours lecture given during the workshop "Metastability and Stochastic Processes" held in Marne-la-Vallée in September 21st-23rd 2011. I would like to warmly thank the organizers Tony Lelièvre and Arnaud Guillin for a very nice organization and for obliging me first to give the lecture, second to write these notes. I also want to acknowledge all the people who attended the lecture.Résumé. Ces notes correspondent à un cours de trois heures donné durant le workshop "Metastability and Stochastic Processes" organisé à Marne-la-Vallée les 21-23 septembre 2011.

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Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems

Cited by 172 publications

References 17 publications

Transportation-information inequalities for Markov processes

Transportation-information inequalities for Markov processes

Large deviations results for the stochastic Navier–Stokes equations

Long time behavior of Markov processes

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