Results 1  10
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27
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Transportationinformation inequalities for Markov processes (II): Relations with other functional inequalities
, 2009
"... We continue our investigation on the transportationinformation inequalities WpI for a symmetric markov process, introduced and studied in [13]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure µ. We giv ..."
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Cited by 31 (10 self)
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We continue our investigation on the transportationinformation inequalities WpI for a symmetric markov process, introduced and studied in [13]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure µ. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion process implies W1I (a result due to [13]) and a Cheeger type’s isoperimetric inequality. Finally we exhibit relations between transportationinformation inequalities and a family of functional inequalities (such as Φlog Sobolev or ΦSobolev).
A CHARACTERIZATION OF DIMENSION FREE CONCENTRATION IN TERMS OF TRANSPORTATION INEQUALITIES
, 2008
"... The aim of this paper is to show that a probability measure µ on R d concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand’s T2 transportationcost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villan ..."
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Cited by 18 (10 self)
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The aim of this paper is to show that a probability measure µ on R d concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand’s T2 transportationcost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincaré inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.
Gradient estimates and the first Neumann eigenvalue on nonconvex manifolds
 Stoch. Proc. Appl
"... By studying the local time of reflecting diffusion processes, explicit gradient estimates of the Neumann heat semigroup on nonconvex manifolds are derived from a recent derivative formula established by Hsu. As an application, an explicit lower bound of the first Neumann eigenvalue is presented via ..."
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Cited by 16 (9 self)
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By studying the local time of reflecting diffusion processes, explicit gradient estimates of the Neumann heat semigroup on nonconvex manifolds are derived from a recent derivative formula established by Hsu. As an application, an explicit lower bound of the first Neumann eigenvalue is presented via dimension, radius and bounds of the curvature and the second fundamental form. Finally, some new estimates are also presented for the strictly convex case.
HamiltonJacobi equations on metric spaces and transport entropy inequalities
, 2012
"... Abstract. We prove an HopfLaxOleinik formula for the solutions of some HamiltonJacobi equations on a general metric space. As a first consequence, we show in full generality that the logSobolev inequality is equivalent to an hypercontractivity property of the HamiltonJacobi semigroup. Asasecon ..."
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Cited by 13 (4 self)
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Abstract. We prove an HopfLaxOleinik formula for the solutions of some HamiltonJacobi equations on a general metric space. As a first consequence, we show in full generality that the logSobolev inequality is equivalent to an hypercontractivity property of the HamiltonJacobi semigroup. Asasecondconsequence, weprovethatTalagrand’s transportentropy inequalities in metric space are characterized in terms of logSobolev inequalities restricted to the class of cconvex functions. hal00795829, version 1 1 Mar 2013 1.
Poincaré inequalities and dimension free concentration of measure
 Ann. Inst. Henri Poincaré Probab. Stat
"... Abstract. In this paper, we consider Poincaré inequalities for nonEuclidean metrics on Rd. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and G ..."
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Cited by 13 (7 self)
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Abstract. In this paper, we consider Poincaré inequalities for nonEuclidean metrics on Rd. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and Gaussian and beyond. We give equivalent functional forms of these Poincaré type inequalities in terms of transportationcost inequalities and infconvolution inequalities. Workable sufficient conditions are given and a comparison is made with super Poincaré inequalities. Résumé. Dans cet article, nous introduisons des inégalités de Poincaré pour des métriques noneuclidiennes sur Rd et nous montrons qu’elles entraînent des inégalités de concentrations adimensionnelles pour les mesures produits. Cette technique nous permet d’atteindre un spectre très large de taux de concentration, aussi bien sous et surgaussiens. Par ailleurs, nous montrons que ces inégalités de Poincaré admettent des formes fonctionnelles équivalentes en termes d’inégalités de transport et d’infconvolution. Enfin, nous donnons des conditions suffisantes pour ces inégalités de Poincaré et nous les comparons aux inégalités superPoincaré. MSC: 60E15; 26D10
TransportationCost Inequalities on Path Space Over Manifolds with Boundary ∗
, 2009
"... Let L = ∆+Z for a C 1 vector field Z on a complete Riemannian manifold possibly with a boundary. By using the uniform distance, a number of transportationcost inequalities on the path space for the (reflecting) Ldiffusion process are proved to be equivalent to the curvature condition Ric − ∇Z ≥ −K ..."
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Cited by 9 (1 self)
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Let L = ∆+Z for a C 1 vector field Z on a complete Riemannian manifold possibly with a boundary. By using the uniform distance, a number of transportationcost inequalities on the path space for the (reflecting) Ldiffusion process are proved to be equivalent to the curvature condition Ric − ∇Z ≥ −K and the convexity of the boundary (if exists). These inequalities are new even for manifolds without boundary, and are partly extended to nonconvex manifolds by using a conformal change of metric which makes the boundary from nonconvex to convex. AMS subject Classification: 60J60, 58G60. Keywords: Transportationcost inequality, curvature, second fundamental form, path space. 1
Measure Concentration, Transportation Cost, And Functional Inequalities
 Summer School on Singular Phenomena and Scaling in Mathematical Models
, 2003
"... In these lectures, we present a triple description of the concentration of measure phenomenon, geometric (through BrunnMinkoswki inequalities), measuretheoretic (through transportation cost inequalities) and functional (through logarithmic Sobolev inequalities), and investigate the relationship ..."
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Cited by 9 (1 self)
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In these lectures, we present a triple description of the concentration of measure phenomenon, geometric (through BrunnMinkoswki inequalities), measuretheoretic (through transportation cost inequalities) and functional (through logarithmic Sobolev inequalities), and investigate the relationships between these various viewpoints. Special emphasis is put on optimal mass transportation and the dual hypercontractive bounds on solutions of HamiltonJacobi equations that o#er a unified treatment of these various aspects.
Properties of Isoperimetric, Functional and TransportEntropy Inequalities Via Concentration
, 2009
"... Various properties of isoperimetric, functional, TransportEntropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure ..."
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Cited by 7 (3 self)
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Various properties of isoperimetric, functional, TransportEntropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a onesided L ∞ bound on the ratio between their densities, Wasserstein distances, and KullbackLeibler divergence. In particular, an extension of the Holley–Stroock perturbation lemma for the logSobolev inequality is obtained. Second, the equivalence of TransportEntropy inequalities with different cost functions is verified, by obtaining a reverse Jensen type inequality. In view of a recent result of Gozlan, this is used to obtain tensorization properties of concentration inequalities with respect to various productmetrics, and the tensorization result for isoperimetric inequalities of Barthe–Cattiaux–Roberto is easily recovered. Some further applications are also described. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting.
From super Poincaré to weighted logSobolev and entropycost inequalities
 J. Math. Pures Appl
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