Results 1  10
of
20
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré
 J. Func. Anal
, 1996
"... Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new ..."
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Cited by 25 (14 self)
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Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (LyapunovPoincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic FokkerPlanck equation recently studied by HérauNier, HelfferNier and Villani is in particular discussed in the final section.
Isoperimetry between exponential and Gaussian
 Electronic J. Prob
"... We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1 ..."
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Cited by 16 (7 self)
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We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1
Logarithmic Sobolev Inequalities and Spectral Gaps
, 2004
"... We prove an simple entropy inequality and apply it to the problem of determining log– Sobolev constants. 1 ..."
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Cited by 7 (0 self)
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We prove an simple entropy inequality and apply it to the problem of determining log– Sobolev constants. 1
QUASISTATIONARY DISTRIBUTIONS AND DIFFUSION MODELS IN POPULATION DYNAMICS
, 2009
"... In this paper, we study quasistationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to − ∞ at the origin, and the diffusion to have an entrance boundary at +∞. These diffusions arise as images, by a deterministic map, of generalized Feller d ..."
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Cited by 6 (3 self)
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In this paper, we study quasistationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to − ∞ at the origin, and the diffusion to have an entrance boundary at +∞. These diffusions arise as images, by a deterministic map, of generalized Feller diffusions, which themselves are obtained as limits of rescaled birth–death processes. Generalized Feller diffusions take nonnegative values and are absorbed at zero in finite time with probability 1. An important example is the logistic Feller diffusion. We give sufficient conditions on the drift near 0 and near + ∞ for the existence of quasistationary distributions, as well as rate of convergence in the Yaglom limit and existence of the Qprocess. We also show that under these conditions, there is exactly one quasistationary distribution, and it attracts all initial distributions under the conditional evolution, if and only if + ∞ is an entrance boundary. In particular this gives a sufficient condition for the uniqueness of quasistationary distributions. In the proofs spectral theory plays an important role on L 2 of the reference measure for the killed process.
A qualitative study of linear driftdiffusion equations with timedependent or vanishing coefficients
, 2005
"... This paper is concerned with entropy methods for linear driftdiffusion equations with explicitly timedependent or degenerate coefficients. Our goal is to establish a list of various qualitative properties of the solutions. The motivation for this study comes from a model for molecular motors, the ..."
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Cited by 3 (0 self)
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This paper is concerned with entropy methods for linear driftdiffusion equations with explicitly timedependent or degenerate coefficients. Our goal is to establish a list of various qualitative properties of the solutions. The motivation for this study comes from a model for molecular motors, the socalled Brownian ratchet, and from a nonlinear equation arising in traffic flow models, for which complex long time dynamics occurs. General results are out of the scope of this paper, but we deal with several examples corresponding to most of the expected behaviors of the solutions. We first prove a contraction property for general entropies which is a useful tool for uniqueness and for the convergence to some eventually timedependent large time asymptotic solutions. Then we focus on power law and logarithmic relative entropies. When the diffusion term is of the type ∇(x  α ∇·), we prove that the inequality relating the entropy with the entropy production term is a HardyPoincaré type inequality, that we establish. Here we assume that α ∈ (0, 2] and the limit case α = 2 appears as a threshold for the method. As a consequence, we obtain an exponential decay of the relative entropies. In the case of timeperiodic coefficients, we prove the existence of a unique timeperiodic solution which attracts all other solutions. The case of a degenerate diffusion coefficient taking the form x  α with α> 2 is also studied. The Gibbs state exhibits a non integrable singularity. In this case concentration phenomena may occur, but we conjecture that an additional timedependence restores the smoothness of the asymptotic solution.
Convex Sobolev inequalities and spectral gap
, 2005
"... This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by ..."
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This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux [11] and Carlen and Loss [10] for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case.
SYLVIE MÉLÉARD ♠ Ecole Polytechnique
"... Competitive or weak cooperative stochastic LotkaVolterra systems conditioned to nonextinction ..."
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Competitive or weak cooperative stochastic LotkaVolterra systems conditioned to nonextinction
AND
, 2003
"... Abstract. We show that the quadratic transportation cost inequality T2 is equivalent to both a Poincaré inequality and a strong form of the Gaussian concentration property. In particular if a logarithmic Sobolev inequality implies T2, we are able to give examples for which T2 holds but the logarithm ..."
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Abstract. We show that the quadratic transportation cost inequality T2 is equivalent to both a Poincaré inequality and a strong form of the Gaussian concentration property. In particular if a logarithmic Sobolev inequality implies T2, we are able to give examples for which T2 holds but the logarithmic Sobolev inequality does not hold. This answers to a question left open by Otto and Villani [21] and Bobkov, Gentil and Ledoux [4], and furnishes (in a Riemannian setting) the analogue of the well known criterion by Bobkov and Götze for the linear transportation cost inequality T1 [5] (also see [12]). The main ingredient in the proof is a new family of inequalities, called modified quadratic transportation cost inequalities in the spirit of the modified logarithmicSobolev inequalities by Bobkov and Ledoux [6], that are shown to hold as soon as a Poincaré inequality is satisfied.