Results 1  10
of
16
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Transportationinformation inequalities for Markov processes (II): Relations . . .
, 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 16 (2 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 7 (4 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.
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 6 (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.
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 3 (1 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 for non euclidean metrics and . . .
, 2007
"... In this paper, we consider Poincaré inequalities for non euclidean metrics on R d. 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 ..."
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Cited by 2 (0 self)
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In this paper, we consider Poincaré inequalities for non euclidean metrics on R d. 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 different equivalent functional forms of these Poincaré type inequalities in terms of transportationcost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized BecknerLatalaOleszkiewicz inequalities.
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 1 (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
Contents
, 2005
"... Abstract. We establish some quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field problem. The tools include coupling arguments, as well ..."
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Abstract. We establish some quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances. As an application, we provide some error bounds for particle simulations in a model mean field problem. The tools include coupling arguments, as well as regularity and moments estimates for solutions of certain diffusive partial differential equations.
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.
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 [20] and Bobkov, Gentil and Ledoux [3], and furnishes (in a Riemannian setting) the analogue of the well known criterion by Bobkov and Götze for the linear transportation cost inequality T1 [4] (also see [11]). 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 [5], that are shown to hold as soon as a Poincaré inequality is satisfied.