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4 results on '"Super Poincaré inequalities"'

1. Poincaré inequalities and dimension free concentration of measure.

Author

Gozlan, Nathael

Subjects
*MATHEMATICAL inequalities, *NON-Euclidean geometry, *MATHEMATICAL convolutions, *GAUSSIAN processes, *PROBABILITY theory, *MATHEMATICAL statistics
Abstract

In this paper, we consider Poincaré inequalities for non-Euclidean 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 transportation-cost inequalities and inf-convolution inequalities. Workable sufficient conditions are given and a comparison is made with super Poincaré inequalities. [ABSTRACT FROM AUTHOR]

Published
2010
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2. Lyapunov conditions for Super Poincaré inequalities

Author

Cattiaux, Patrick, Guillin, Arnaud, Wang, Feng-Yu, and Wu, Liming

Subjects
*LYAPUNOV functions, *POINCARE series, *MATHEMATICAL inequalities, *FUNCTIONAL analysis, *SOBOLEV spaces, *MATHEMATICAL analysis
Abstract

Abstract: We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or F-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in [D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal. 254 (3) (2008) 727–759. Available on Mathematics arXiv:math.PR/0703355, 2007]. This approach allows us to recover and extend in a unified way some known criteria in the euclidean case (Bakry and Emery, Wang, Kusuoka and Stroock, …). [Copyright &y& Elsevier]

Published
2009
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3. Poincaré inequalities and dimension free concentration of measure

Author

Nathael Gozlan, Laboratoire d'Analyse et de Mathématiques Appliquées, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)

Subjects
Statistics and Probability, Pure mathematics, Inequality, media_common.quotation_subject, Inf-convolution inequalities, MathematicsofComputing_NUMERICALANALYSIS, Poincaré inequality, Super Poincaré inequalities, 01 natural sciences, Sobolev inequality, 010104 statistics & probability, symbols.namesake, Dimension (vector space), Probability theory, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, media_common, Mathematics, Concentration of measure, 010102 general mathematics, Transportation-cost inequalities, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Logarithmic-Sobolev inequalities, Poincaré conjecture, symbols, 60E15, Statistics, Probability and Uncertainty, 26D10
Abstract

Dans cet article, nous introduisons des inegalites de Poincare pour des metriques non-euclidiennes sur ℝ d et nous montrons qu'elles entrainent des inegalites de concentrations adimensionnelles pour les mesures produits. Cette technique nous permet d'atteindre un spectre tres large de taux de concentration, aussi bien sous et sur-gaussiens. Par ailleurs, nous montrons que ces inegalites de Poincare admettent des formes fonctionnelles equivalentes en termes d'inegalites de transport et d'inf-convolution. Enfin, nous donnons des conditions suffisantes pour ces inegalites de Poincare et nous les comparons aux inegalites super-Poincare.

Published
2010
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4. Lyapunov conditions for logarithmic Sobolev and Super Poincaré inequality

Author

Cattiaux, Patrick, Guillin, Arnaud, Wang, Feng-Yu, Wu, Liming, Centre de Mathématiques Appliquées (CMAP), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Statistique et Probabilités (LSP), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Analyse, Topologie, Probabilités (LATP), Université Paul Cézanne - Aix-Marseille 3-Université de Provence - Aix-Marseille 1-Centre National de la Recherche Scientifique (CNRS), School of Mathematical Sciences [Beijing] (SMS), Beijing Normal University (BNU), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics Wuhan University, Wuhan University [China], Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)

Subjects
26D10, 47D07, 60G10, 60J60, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Ergodic processes, logarithmic Sobolev inequalities, Mathematics::Metric Geometry, Mathematics::Differential Geometry, Poincaré inequalities, Super Poincaré inequalities, Mathematics - Probability, Lyapunov functions
Abstract

International audience; We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in Bakry and al. This approach allows us to recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock ...).

Published
2009

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