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1. The Li–Yau inequality and applications under a curvature-dimension condition

Author

Dominique Bakry, François Bolley, and Ivan Gentil

Subjects
Pure mathematics, Algebra and Number Theory, Markov chain, Semigroup, 010102 general mathematics, Dimension (graph theory), Type (model theory), Riemannian manifold, Curvature, 01 natural sciences, 010101 applied mathematics, Mathematics::Differential Geometry, Geometry and Topology, 0101 mathematics, Heat kernel, Mathematics, Harnack's inequality
Abstract

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.

Published
2017
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2. The Markov sequence problem for the Jacobi polynomials and on the simplex *

Author

Dominique Bakry, Lamine Mbarki, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Centre National de la Recherche Scientifique ( CNRS ) -Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -PRES Université de Toulouse-Université Paul Sabatier - Toulouse 3 ( UPS ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse 1 Capitole ( UT1 ), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Markov sequence, Simplex, General Mathematics, Dirichlet measures, 010102 general mathematics, 16. Peace & justice, 01 natural sciences, 33C45, Combinatorics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Markov sequences, 010104 statistics & probability, symbols.namesake, 43A90, 43A62, Orthogonal polynomials, hypergroups, symbols, Jacobi polynomials, 46H99, 60J99, 0101 mathematics, orthogonal polynomials, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics
Abstract

The Markov sequence problem aims at the description of possible eigenvalues of symmetric Markov operators with some given orthonormal basis as eigenvector decomposition. A fundamental tool for their description is the hypergroup property. We first present the general Markov sequence problem and provide the classical examples, most of them associated with the classical families of orthogonal polynomials. We then concentrate on the hypergroup property, and provide a general method to obtain it, inspired by a fundamental work of Carlen, Geronimo and Loss. Using this technique and a few properties of diffusion operators having polynomial eigenvectors, we then provide a simplified proof of the hypergroup property for the Jacobi polynomials (Gasper’s theorem) on the unit interval. We finally investigate various generalizations of this property for the family of Dirichlet laws on the simplex.

Published
2018

3. On Harnack inequalities and optimal transportation

Author

Dominique Bakry, Ivan Gentil, and Michel Ledoux

Subjects
Inequality, media_common.quotation_subject, Curvature, Theoretical Computer Science, Mathematics (miscellaneous), Mathematics::Probability, Bounded function, Applied mathematics, Mathematics::Differential Geometry, Contraction (operator theory), Heat flow, Heat kernel, Harnack's inequality, Mathematics, media_common
Abstract

We develop connections between Harnack inequalities for the heat flow of diffusion operators with curvature bounded from below and optimal transportation. Through heat kernel inequalities, a new isoperimetric-type Harnack inequality is emphasized. Commutation properties between the heat and Hopf-Lax semigroups are developed consequently, providing direct access to the heat flow contraction property along Wasserstein distances.

Published
2015
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4. Sharp Beckner-type inequalities for Cauchy and spherical distributions

Author

Ivan Gentil, Dominique Bakry, Grégory Scheffer, Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Camille Jordan [Villeurbanne] ( ICJ ), Centre National de la Recherche Scientifique ( CNRS ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Équations aux dérivées partielles, analyse (EDPA), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Équations aux dérivées partielles, analyse ( EDPA ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ) -École Centrale de Lyon ( ECL ), and Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Beckner inequality, Pure mathematics, General Mathematics, Stochastic calculus, Poincaré inequality, Harmonic (mathematics), Stopping time, [ MATH.MATH-FA ] Mathematics [math]/Functional Analysis [math.FA], Type (model theory), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Bessel processes, Sobolev inequality, 010104 statistics & probability, symbols.namesake, Cauchy distribution, FOS: Mathematics, 0101 mathematics, Mathematics, curvature-dimension condition, Euclidean space, Plane (geometry), 010102 general mathematics, Probability (math.PR), Stochastic calculus Mathematics Subject Classification (2010): 60G10, Spherical analysis, Functional Analysis (math.FA), Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 58-XX, 60-XX, symbols, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability
Abstract

International audience; Using some harmonic extensions on the upper-half plane, and probabilistic representations, and curvature-dimension inequalities with some negative dimensions, we obtain some new opimal functional inequalities of the Beckner type for the Cauchy type distributions on the Euclidean space. These optimal inequalities appear to be equivalent to some non tight optimal Beckner inequalities on the sphere, and the family appear to be a new form of the Sobolev inequality.

Published
2018
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5. Harnack inequalities on a manifold with positive or negative Ricci curvature

Author

Zhongmin Qian and Dominique Bakry

Subjects
Riemann curvature tensor, General Mathematics, Mathematical analysis, Ricci flow, Riemannian manifold, symbols.namesake, Harnack's principle, symbols, Mathematics::Differential Geometry, Heat kernel, Ricci curvature, Scalar curvature, Mathematics, Harnack's inequality
Abstract

Several new Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established. These estimates are sharp both for small time, for large time and for large distance, and lead to new estimates for the heat kernel of a manifold with Ricci curvature bounded below.

Published
2016

6. Preface

Author

Dominique Bakry, Serge Cohen, Martin Hairer, and Jean-Michel Roquejoffre

Subjects
General Medicine
Published
2017
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7. Dimension dependent hypercontractivity for Gaussian kernels

Author

François Bolley, Dominique Bakry, Ivan Gentil, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-08-BLAN-0242,EVOL,Dissipative Evolutions and Convergence to Equilibrium(2008), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Large class, Pure mathematics, Trace (linear algebra), Markov kernel, Gaussian, Logarithmic Sobolev inequality, Transportation inequality, 01 natural sciences, 010104 statistics & probability, symbols.namesake, Mathematics::Probability, Dimension (vector space), Curvature-dimension criterion, FOS: Mathematics, Hypercontractive bound, 0101 mathematics, Mathematics, Diffusion semigroup, Markov chain, Semigroup, Probability (math.PR), 010102 general mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], symbols, Statistics, Probability and Uncertainty, Mathematics - Probability, Analysis
Abstract

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck semigroup driven by a non-diffusive L\'evy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis., Comment: 24 pages

Published
2011
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8. Around Nash inequalities

Author

Ivan Gentil, François Bolley, and Dominique Bakry

Subjects
TheoryofComputation_MISCELLANEOUS, Computer Science::Computer Science and Game Theory, Inequality, media_common.quotation_subject, TheoryofComputation_GENERAL, General Medicine, symbols.namesake, Simple (abstract algebra), Nash equilibrium, symbols, Economics, Price of stability, Mathematical economics, media_common
Abstract

The aim of this short note is to explain how Nash inequalities lead to such estimates in a general setting and also to show simple techniques used to establish the required Nash inequalities.

Published
2010
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9. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré

Author

Patrick Cattiaux, Dominique Bakry, Arnaud Guillin, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), 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), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Lyapunov function, 010102 general mathematics, Mathematical analysis, Markov process, 60J35 (47D07 60J25), Lyapunov exponent, Type (model theory), Poincaré inequalities, 01 natural sciences, 010104 statistics & probability, symbols.namesake, Hypocoercivity, Rate of convergence, Ergodic processes, Tweedie distribution, symbols, Ergodic theory, Applied mathematics, Lyapunov equation, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Analysis, Mathematics, Lyapunov functions
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 (Lyapunov–Poincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker–Planck equation recently studied by Hérau and Nier, Helffer and Nier, and Villani is in particular discussed in the final section.

Published
2008
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10. Analysis and Geometry of Markov Diffusion Operators

Author

Dominique Bakry, Ivan Gentil, Michel Ledoux, Dominique Bakry, Ivan Gentil, and Michel Ledoux

Subjects
Markov operators, Distribution (Probability theory)
Abstract

The present volume is an extensive monograph on the analytic and geometric aspects of Markov diffusion operators. It focuses on the geometric curvature properties of the underlying structure in order to study convergence to equilibrium, spectral bounds, functional inequalities such as Poincaré, Sobolev or logarithmic Sobolev inequalities, and various bounds on solutions of evolution equations. At the same time, it covers a large class of evolution and partial differential equations. The book is intended to serve as an introduction to the subject and to be accessible for beginning and advanced scientists and non-specialists. Simultaneously, it covers a wide range of results and techniques from the early developments in the mid-eighties to the latest achievements. As such, students and researchers interested in the modern aspects of Markov diffusion operators and semigroups and their connections to analytic functional inequalities, probabilistic convergence to equilibrium andgeometric curvature will find it especially useful. Selected chapters can also be used for advanced courses on the topic.

Published
2014

11. Curvature dimension bounds on the deltoid model

Author

Dominique Bakry, Olfa Zribi, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Probability (math.PR), 010102 general mathematics, Boundary (topology), General Medicine, Curvature, 01 natural sciences, Dirichlet distribution, Sobolev inequality, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], symbols.namesake, Dimension (vector space), Bounded function, 0103 physical sciences, Orthogonal polynomials, FOS: Mathematics, symbols, 010307 mathematical physics, 0101 mathematics, Mathematics - Probability, Mathematics, Probability measure
Abstract

International audience; The deltoid curve in R 2 is the boundary of a domain on which there exist probability measures and orthogonal polynomials for theses measures which are eigenvec-tors of diffusion operators. As such, they may be considered as a two dimensional extension of the classical Jacobi operators. They belong to one of the 11 families of such bounded domains in R 2. We study the curvature-dimension inequalities associated to these operators, and deduce various bounds on the associated polynomials, together with Sobolev inequalities related to the associated Dirichlet forms

Published
2016
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12. A logarithmic Sobolev form of the Li-Yau parabolic inequality

Author

Dominique Bakry, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
60H, 60J, General Mathematics, non-negative curvature, Poincaré inequality, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, heat semigroup, gradient estimate, Sobolev inequality, Li-Yau inequality, 010104 statistics & probability, symbols.namesake, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Log sum inequality, diameter bound, 0101 mathematics, logarithmic Sobolev inequality, Heat kernel, Mathematics, Markov semigroups, heat equation, Euclidean space, 010102 general mathematics, Mathematical analysis, crvature-dimension inequalities, Gaussian measure, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Sobolev space, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Li-Yau parabolic inequality, symbols, Heat equation, 58J, diffusion semigroups
Abstract

International audience; We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities in this setting. Exponential Laplace differential inequalities through the Herbst argument furthermore yield diameter bounds and dimensional estimates on the heat kernel volume of balls.

Published
2006
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13. Extension of Bochner-Lichnérowicz formula on spheres

Author

Dominique Bakry and Abdellatif Bentaleb

Subjects
010104 statistics & probability, 010102 general mathematics, Lichnerowicz formula, Geometry, General Medicine, Extension (predicate logic), 0101 mathematics, 01 natural sciences, Humanities, Mathematics
Abstract

On definit l'espace des formes bilineaires intrinseques associees a un operateur L differentiel du second ordre. Ces formes sont definies uniquement a partir de l'operateur L lui-meme et de la structure d'algebre sur les fonctions donnee par la multiplication. La positivite d'une de ces formes ne depend que de la structure locale de l'operateur. Pour des operateurs symetriques par rapport a une mesure positive, chaque forme bilineaire intrinseque positive donne des informations sur le spectre de L. Dans cet article, nous construisons une famille (R k ) de formes bilineaires intrinseques sur les spheres (dans ce cas, l'operateur L est l'operateur de Laplace-Beltrami) qui contient toute l'information sur le spectre. Plus precisement, si f est un vecteur propre de l'operateur de Laplace-Beltrami de valeur propre λ, et si g est une fonction lisse quelconque, alors pour la mesure Riemannienne μ, on a ∫R k (f,g)dμ = λ(λ-λ 1 )...(λ-λ k-1 ) ∫ fgdμ, ou 0, λ 1 ,...,λ k-1 sont les k premieres valeurs propres de l'operateur de Laplace-Beltrami. C'est une extension a tout le spectre de la formule de Bochner-Lichnerowicz, qui donne une minoration precise de la premiere valeur propre non nulle. On etend ensuite cette propriete a tout une famille d'operateurs du second ordre qui est une extension multidimensionnelle des operateurs ultraspheriques en dimension 1.

Published
2005
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14. Hypergroup properties for the deltoid model

Author

Olfa Zribi, Dominique Bakry, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Property (philosophy), Plane (geometry), Orthogonal polynomials, General Mathematics, 010102 general mathematics, Deltoid curve, Probability (math.PR), diffusion processes, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], 01 natural sciences, 33C52, 31C35, 35K05, 60J60, 60J45, hypergroup, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Classical Analysis and ODEs, 0103 physical sciences, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 010307 mathematical physics, 0101 mathematics, Mathematics - Probability, Mathematics
Abstract

International audience; We investigate the hypergroup property for the family of orthogonal polynomials associated with the deltoid curve in the plane, related to the A 2 root system. This provides also the same property for another family of polynomials related to the G 2 root system.

Published
2015
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15. Diffusions with polynomial eigenvectors via finite subgroups of O(3)

Author

Dominique Bakry, Xavier Bressaud, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Polynomial, Pure mathematics, Regular polyhedron, 42C05, regular polyhedra, Orthogonal polynomials, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], diffusion operators, Commutative Algebra (math.AC), 01 natural sciences, 010104 statistics & probability, FOS: Mathematics, 0101 mathematics, Invariant (mathematics), 47D07, Eigenvalues and eigenvectors, 33C590, Mathematics, 010102 general mathematics, Probability (math.PR), diffusion processes, General Medicine, Mathematics - Commutative Algebra, invariant polynomials MSC classification: 13A50, 58J65, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Mathematics - Probability
Abstract

International audience; We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction rely on the finite subgroups of O(3) and their invariant polynomials.

Published
2015
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16. h-Transforms and Orthogonal Polynomials

Author

Olfa Zribi and Dominique Bakry

Subjects
Gegenbauer polynomials, Discrete orthogonal polynomials, 010102 general mathematics, 01 natural sciences, Algebra, Classical orthogonal polynomials, symbols.namesake, 0103 physical sciences, Wilson polynomials, Orthogonal polynomials, Hahn polynomials, symbols, Jacobi polynomials, 010307 mathematical physics, 0101 mathematics, Koornwinder polynomials, Mathematics
Abstract

We describe some examples of classical and explicit h-transforms as particular cases of a general mechanism, which is related to the existence of symmetric diffusion operators having orthogonal polynomials as spectral decomposition.

Published
2015
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17. Some New Results on Eigenvectors via Dimension, Diameter, and Ricci Curvature

Author

Zhongmin Qian and Dominique Bakry

Subjects
Mathematics(all), Elliptic operator, Curvature of Riemannian manifolds, General Mathematics, Bounded function, Mathematical analysis, Spectral gap, Mathematics::Differential Geometry, Upper and lower bounds, Eigenvalues and eigenvectors, Ricci curvature, Scalar curvature, Mathematics
Abstract

We generalise for a general symmetric elliptic operator the different notions of dimension, diameter, and Ricci curvature, which coincide with the usual notions in the case of the Laplace-Beltrami operators on Riemannian manifolds. If λ 1 denotes the spectral gap, that is the first nonzero eigenvalue, we investigate in this paper the best lower bound on λ 1 one can obtain under an upper bound on the dimension, an upper bound on the diameter, and a lower bound of the Ricci curvature. Two cases are known: namely if the Ricci curvature is bounded below by a constant R>0, then λ 1≥nR/(n-1), and this estimate is sharp for the n-dimensional spheres (Lichnerowicz's bound). If the Ricci curvature is bounded below by zero, then Zhong-Yang's estimate asserts that λ 1≥π 2d 2, where d is an upper bound on the diameter. This estimate is sharp for the 1-dimensional torus. In the general case, many interesting estimates have been obtained. This paper provides a general optimal comparison result for λ 1 which unifies and sharpens Lichnerowicz and Zhong-Yang's estimates, together with other comparison results concerning the range of the associated eigenfunctions and their derivatives. © 2000 Academic Press.

Published
2000
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19. Dyson Processes Associated with Associative Algebras: The Clifford Case

Author

Dominique Bakry and Marguerite Zani

Subjects
Pure mathematics, Mathematics::Probability, Clifford algebra, Symmetric matrix, Invariant measure, Laplace operator, Associative property, Eigenvalues and eigenvectors, Brownian motion, Characteristic polynomial, Mathematics
Abstract

We consider Brownian motions and other processes (Ornstein-Uhlenbeck processes, spherical Brownian motions) on various sets of symmetric matrices constructed from algebra structures, and look at their associated spectral measure processes. This leads to the identification of the multiplicity of the eigenvalues, together with the identification of the spectral measures. For Clifford algebras, we thus recover Bott’s periodicity.

Published
2014
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20. Symmetric diffusions with polynomial eigenvectors

Author

Dominique Bakry, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Crisan D., Hambly B., Zariphopoulou T. (eds), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
60B15, 60B20, 60G99, 43A75, 14H50, Power sum symmetric polynomial, Discrete orthogonal polynomials, 010102 general mathematics, diffusion operators, Complete homogeneous symmetric polynomial, 01 natural sciences, random matrices, 010101 applied mathematics, Classical orthogonal polynomials, Combinatorics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], symbols.namesake, Orthogonal polynomials, symbols, Jacobi polynomials, Elementary symmetric polynomial, 0101 mathematics, Ring of symmetric functions, orthogonal polynomials, Mathematics
Abstract

25 pages; International audience; We describe symmetric diffusion operators where the spectral decomposition is given through a family of orthogonal polynomials. In dimension one, this reduces to the case of Hermite, Laguerre and Jacobi polynomials. In higher dimension, some basic examples arise from compact Lie groups. We give a complete description of the bounded sets on which such operators may live. We then provide a classification of those sets when the polynomials are ordered according to their usual degrees

Published
2014
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21. Optimal Transportation and Functional Inequalities

Author

Michel Ledoux, Dominique Bakry, and Ivan Gentil

Subjects
Sobolev space, Geodesic, Semigroup, Euclidean space, Applied mathematics, Contraction (operator theory), Ricci curvature, Sobolev inequality, Probability measure, Mathematics
Abstract

This chapter is a brief investigation of the links between optimal transportation methods and functional inequalities in the Markov operator framework of this monograph. After a brief introduction to the basic material on optimal transportation, the main topic of transportation cost inequalities and first examples for Gaussian measures are presented. Interpolation along the geodesics of optimal transport is used towards logarithmic Sobolev inequalities and transportation cost inequalities comparing relative entropy and Wasserstein distances between probability measures. An alternate approach to sharp Sobolev or Gagliardo–Nirenberg inequalities in Euclidean space is provided next along these lines. Non-linear Hamilton–Jacobi equations and hypercontractivity properties of their solutions, analogous to the ones for linear heat equations, are investigated in the further sections towards the relationships between (quadratic) transportation cost inequalities and logarithmic Sobolev inequalities. Contraction properties in Wasserstein space along with the heat semigroup are investigated in the Markov operator setting. The last section is a very brief overview of recent developments towards a notion of Ricci curvature lower bounds based on optimal transportation and the connection with the Γ-calculus developed in this work.

Published
2014
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22. Logarithmic Sobolev Inequalities

Author

Ivan Gentil, Dominique Bakry, and Michel Ledoux

Subjects
Mathematics::Functional Analysis, Pure mathematics, Logarithm, Semigroup, Mathematics::Analysis of PDEs, Invariant measure, Gaussian measure, Lipschitz continuity, Real line, Heat kernel, Mathematics, Sobolev inequality
Abstract

After Poincare inequalities, logarithmic Sobolev inequalities are amongst the most studied functional inequalities for semigroups. They contain much more information than Poincare inequalities, and are at the same time sufficiently general to be available in numerous cases of interest, in particular in infinite dimension (as limits of Sobolev inequalities on finite-dimensional spaces). After the basic definition of a logarithmic Sobolev inequality together with its first properties, the first sections of this chapter present the exponential decay in entropy and the fundamental equivalence between the logarithmic Sobolev inequality and smoothing properties of the semigroup in the form of hypercontractivity. Next, integrability properties of eigenvectors and of Lipschitz functions under a logarithmic Sobolev inequality are discussed together with a criterion for measures on the real line to satisfy a logarithmic Sobolev inequality (for the usual gradient). The further sections deal with curvature conditions, first for the local logarithmic Sobolev inequalities for heat kernel measures, then for the invariant measure with an additional dimensional information. Local hypercontractivity and some applications of the local logarithmic Sobolev inequalities towards heat kernel bounds are further presented. Harnack-type inequalities under the infinite-dimensional curvature conditions, linked with reverse local logarithmic Sobolev inequalities complete the chapter.

Published
2014
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24. Capacity and Isoperimetric-Type Inequalities

Author

Dominique Bakry, Michel Ledoux, and Ivan Gentil

Subjects
Pure mathematics, Dirichlet form, Gaussian isoperimetric inequality, Boundary (topology), Isoperimetric inequality, Gaussian measure, Heat kernel, Sobolev inequality, Mathematics, Harnack's inequality
Abstract

This chapter focuses on inequalities comparing measure and capacity uniformly over a given class of sets as equivalent forms of functional inequalities. The chapter concentrates on the so-called 2-capacities which capture the relevant Dirichlet form information on sets and on the 1-capacities which are related to boundary or surface measures and isoperimetric-type inequalities. The first part of the chapter introduces the basic notions on capacities and co-area formulas to transfer (and back) functional inequalities into measure-capacity inequalities. Sobolev-type, Poincare and logarithmic Sobolev inequalities are analyzed in this respect. The second part is concerned with measure-capacity inequalities. Together with the heat kernel tools under curvature conditions, measure-capacity inequalities of isoperimetric-type are investigated, leading in particular to the Gaussian isoperimetric inequality as well as to comparison results under curvature conditions. Combination of the heat kernel isoperimetric inequality together with its reverse form further yields an isoperimetric-type Harnack inequality. The last section addresses the relationships between Poincare and logarithmic Sobolev inequalities and their associated concentration properties.

Published
2014
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26. Symmetric Markov Diffusion Operators

Author

Dominique Bakry, Michel Ledoux, and Ivan Gentil

Subjects
Algebra, Markov kernel, Markov chain mixing time, Markov chain, Computer science, Dirichlet form, Balance equation, Markov property, Examples of Markov chains, Manifold
Abstract

This chapter provides a general framework for the investigation of symmetric Markov diffusion semigroups and operators. Based on the early observations of the first chapter and on the investigation of the model examples, it develops all the fundamental properties which will justify the computations in the following chapters in the most convenient framework. The main setting consisting of a Markov Triple (E,μ,Γ) describes a convenient framework to develop the formalism of Markov semigroups and the Γ-calculus towards functional inequalities and convergence to equilibrium. The analysis of the concrete example of second order differential operators on smooth complete connected manifolds is the guide for the description, in this framework, of some main features such as connexity, completeness, weak hypo-ellipticity etc. With the main tool of essential self-adjointness at the center of the construction, the chapter emphasizes the relevant hypotheses and properties of Full Markov Triples, in force throughout the monograph, covering the main examples of illustrations.

Published
2014
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27. Generalized Functional Inequalities

Author

Ivan Gentil, Michel Ledoux, and Dominique Bakry

Subjects
Sobolev space, Pure mathematics, Basis (linear algebra), Semigroup, Dirichlet form, Euclidean space, Invariant measure, Heat kernel, Mathematics, Sobolev inequality
Abstract

The Poincare, logarithmic Sobolev and Sobolev inequalities capture different features of the associated semigroup or the invariant measure, in terms of convergence to equilibrium, estimates on the heat kernels or tail behaviors of the invariant measure. This chapter investigates intermediate or more general families of functional inequalities which are suited to a wide variety of regimes as well as to more precise, or different, features. The study is concerned with three main examples, entropy-energy, generalized Nash and weak Poincare inequalities. The first part of the chapter describes the family of functional entropy–energy inequalities well-suited to heat kernel bounds by means of the method developed for hypercontractivity under logarithmic Sobolev inequalities. Off-diagonal heat kernel estimates may be achieved in the same way. Further sections in this chapter investigate generalized Nash inequalities on the basis of the example of the classical Nash inequality in Euclidean space, as well as weighted Nash inequalities, with a focus on non-uniform heat kernel bounds and tail inequalities. Weak Poincare inequalities are studied as the main minimal tool to tighten families of standard functional inequalities. Weak Poincare inequalities may be further studied in their own from the viewpoint of heat kernel bounds and tail estimates. Further related families of functional inequalities of interest and illustrative examples complete the exposition of this chapter.

Published
2014
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29. Statistique de chaînes de Markov cachées à espace d'états fini. Le cas non stationnaire

Author

Pierre Vandekerkhove, Dominique Bakry, and Xavier Milhaud

Subjects
General Medicine, Humanities, Mathematics
Abstract

Resume L'objet de cette Note est d'etudier, sous des conditions simples, la normalite asymptotique locale du modele statistique associe a une chaine de Markov cachee homogene, a espace fini d'etats, de condition initiale deterministe, ainsi que la consistance et la normalite asymptotique de l'estimateur du maximum de vraisemblance.

Published
1997
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30. On Harnack estimates for positive solutions of the heat equation on a complete manifold

Author

Dominique Bakry and Zhongmin Qian

Subjects
Partial differential equation, Harnack's principle, Differential geometry, Bounded function, Mathematical analysis, Heat equation, Mathematics::Differential Geometry, General Medicine, Riemannian manifold, Ricci curvature, Mathematics, Harnack's inequality
Abstract

We establish several new Harnack estimates for the nonnegative solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded by a positive or negative constant. This extends to symmetric diffusions whose generator satisfies a “curvature-dimension” inequality.

Published
1997
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31. Optimal heat kernel bounds under logarithmic Sobolev inequalities

Author

Daniel Concordet, Michel Ledoux, and Dominique Bakry

Subjects
Statistics and Probability, symbols.namesake, Logarithm, Harmonic function, Euclidean space, Mathematical analysis, symbols, Poincaré inequality, Type (model theory), Laplace operator, Heat kernel, Sobolev inequality, Mathematics
Abstract

We establish optimal uniform upper estimates on heat kernels whose generators satisfy a logarithmic Sobolev inequality (or entropy-energy inequality) with the optimal constant of the Euclidean space. Off-diagonals estimates may also be obtained with however a smaller d istance involving harmonic functions. In the last part, we apply these methods to study some heat kernel decays for diffusion operators of the type Laplacian minus the gradient of a smooth potential with a given growth at infinity.

Published
1997
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32. Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator

Author

Dominique Bakry and Michel Ledoux

Subjects
Hölder's inequality, Loomis–Whitney inequality, General Mathematics, Mathematical analysis, Gaussian isoperimetric inequality, Poincaré inequality, Isoperimetric dimension, Minkowski inequality, Mathematics::Group Theory, symbols.namesake, symbols, Rayleigh–Faber–Krahn inequality, Mathematics::Metric Geometry, Mathematics::Differential Geometry, Isoperimetric inequality, Mathematics
Abstract

We establish, by simple semigroup arguments, a Levy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature.

Published
1996
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33. Weighted Nash Inequalities

Author

Ivan Gentil, Dominique Bakry, François Bolley, Patrick Maheux, Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS), ANR-08-BLAN-0242,EVOL,Dissipative Evolutions and Convergence to Equilibrium(2008), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO)

Subjects
Pure mathematics, General method, Markov chain, General Mathematics, 010102 general mathematics, Mathematical analysis, Probability (math.PR), 01 natural sciences, Sobolev inequality, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Norm (mathematics), FOS: Mathematics, 0101 mathematics, Heat kernel, Mathematics - Probability, Mathematics
Abstract

International audience; Nash or Sobolev inequalities are known to be equivalent to ultracontractive properties of Markov semigroups, hence to uniform bounds on their kernel densities. In this work we present a simple and extremely general method, based on weighted Nash inequalities, to obtain non-uniform bounds on the kernel densities. Such bounds imply a control on the trace or the Hilbert-Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on $\dR$ naturally associated with the measure with density $C_a\exp(-|x|^a)$, with $1

Published
2012
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34. The Hypergroup Property and Representation of Markov Kernels

Author

Nolwen Huet and Dominique Bakry

Subjects
Discrete mathematics, Mathematics::Functional Analysis, Property (philosophy), Markov chain, 010102 general mathematics, Representation (systemics), 16. Peace & justice, Space (mathematics), 01 natural sciences, 010101 applied mathematics, Orthonormal basis, 0101 mathematics, Laplace operator, Eigenvalues and eigenvectors, Probability measure, Mathematics
Abstract

For a given orthonormal basis (f n ) on a probability measure space, we want to describe all Markov operators which have the f n as eigenvectors. We introduce for that what we call the hypergroup property. We study this property in three different cases.

Published
2008
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35. A simple proof of the Poincaré inequality for a large class of probability measures

Author

Dominique Bakry, Franck Barthe, Arnaud Guillin, and Patrick Cattiaux

Subjects
Statistics and Probability, Lyapunov function, Discrete mathematics, Poincaré inequality, log-concave measure, symbols.namesake, Simple (abstract algebra), symbols, Spectral gap, Log sum inequality, Direct proof, 47D07, Statistics, Probability and Uncertainty, Completeness (statistics), 60G10, 26D10, Lyapunov functions, 60J60, Mathematics, Probability measure
Abstract

We give a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on $\mathbb{R}^n$. The proof is based on arguments introduced in Bakry and al, but for the sake of completeness, all details are provided.

Published
2008
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37. On gradient bounds for the heat kernel on the Heisenberg group

Author

Michel Bonnefont, Dominique Bakry, Fabrice Baudoin, Djalil Chafaï, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Physiopathologie et Toxicologie Expérimentales (UPTE), Institut National de la Recherche Agronomique (INRA)-Ecole Nationale Vétérinaire de Toulouse (ENVT), Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, ANR-08-BLAN-0242,EVOL,Dissipative Evolutions and Convergence to Equilibrium(2008), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université de Toulouse (UT), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Ecole Nationale Vétérinaire de Toulouse (ENVT), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National de la Recherche Agronomique (INRA), and ANR-09-EVOL,ANR-09-EVOL

Subjects
Pure mathematics, Type (model theory), Space (mathematics), 01 natural sciences, Heisenberg group, 010104 statistics & probability, Simple (abstract algebra), FOS: Mathematics, functional inequalities, 0101 mathematics, Brownian motion, Heat kernel, Mathematics, 60J60, Probability (math.PR), 010102 general mathematics, Mathematical analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], hypoelliptic diffusions, Hypoelliptic operator, Isoperimetric inequality, 22E30, Analysis, Mathematics - Probability
Abstract

It is known that the couple formed by the two dimensional Brownian motion and its L\'evy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel., Comment: Minor corrections

Published
2007
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38. Perturbations of functional inequalities using growth conditions

Author

Dominique Bakry, Feng-Yu Wang, Michel Ledoux, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), School of Mathematical Sciences [Beijing] (SMS), Beijing Normal University (BNU), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-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 Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics(all), Pure mathematics, Inequality, Applied Mathematics, General Mathematics, media_common.quotation_subject, Logarithmic Sobolev inequality, 010102 general mathematics, Mathematical analysis, Poincaré inequality, 60J45 (58J65 60E15), 01 natural sciences, Perturbation, 010104 statistics & probability, symbols.namesake, symbols, 0101 mathematics, Reference function, ComputingMilieux_MISCELLANEOUS, Logarithmic sobolev inequality, Mathematics, media_common
Abstract

Perturbations of functional inequalities are studied by using merely growth conditions in terms of a distance-like reference function. As a result, optimal sufficient conditions are obtained for perturbations to reach a class of functional inequalities interpolating between the Poincaré inequality and the logarithmic Sobolev inequality.

Published
2007
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39. Characterization of Markov semigroups on ℝ Associated to Some Families of Orthogonal Polynomials

Author

Dominique Bakry and Olivier Mazet

Subjects
Classical orthogonal polynomials, Pure mathematics, symbols.namesake, Hermite polynomials, Markov chain, Discrete orthogonal polynomials, Orthogonal polynomials, Mathematics::Classical Analysis and ODEs, Laguerre polynomials, symbols, Jacobi polynomials, Markov process, Mathematics
Abstract

We give a characterization of the eigenvalues of Markov operators which admit an orthogonal polynomial basis as eigenfunctions, in the Hermite and the Laguerre cases, as well as for the sequences of orthogonal polynomials associated to some probability measures on ℕ. In the Hermite case, we also give a description of the path of the associated Markov processes, as well as a geometric interpretation.

Published
2003
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41. Sur les inégalités GKS

Author

Mireille Echerbault and Dominique Bakry

Abstract

Nous decrivons des bases orthonormees sur l’espace L2 d’un ensemble fini dans lequel des inegalites de la forme GKS1 prennent une forme simple. Ceci s’applique en particulier aux classes de conjugaison d’un groupe fini quelconque. Dans tous les exemples que nous considerons, l’inegalite GKS2 en est une consequence, sans que nous ayons une interpretation generale de ce phenomene.

Published
1996
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45. Weak Sobolev Inequalities

Author

Dominique Bakry

Subjects
Sobolev space, Pure mathematics, Semigroup, Dirichlet form, Mathematics::Analysis of PDEs, Spectral gap, Upper and lower bounds, Heat kernel, Manifold, Mathematics, Sobolev inequality
Abstract

A weak Sobolev inequality (WSI) is a weakened form of a Sobolev inequality associated to a Dirichlet form. It turns out that it is in fact equivalent to a Sobolev inequality. If there is a spectral gap and a (WSI) holds, then we can get a tight weake Sobolev quality (TWSI). Starting with a TWSI. we get upper and lower bounds on the density of the heat semigroup associated with the Dirichlet form, when t → 0 as well as when t → ∞. In the last chapter, we give a Γ2 criterium to get a TWSI on a manifold.

Published
1991
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47. Lectures on Probability Theory : Ecole D'Ete De Probabilites De Saint-Flour XXII - 1992

Author

Dominique Bakry, Richard D. Gill, Stanislav A. Molchanov, Pierre Bernard, Dominique Bakry, Richard D. Gill, Stanislav A. Molchanov, and Pierre Bernard

Subjects
Probabilities
Abstract

This book contains work-outs of the notes of three 15-hour courses of lectures which constitute surveys on the concerned topics given at the St. Flour Probability Summer School in July 1992. The first course, by D. Bakry, is concerned with hypercontractivity properties and their use in semi-group theory, namely Sobolev and Log Sobolev inequa- lities, with estimations on the density of the semi-groups. The second one, by R.D. Gill, is about statistics on survi- val analysis; it includes product-integral theory, Kaplan- Meier estimators, and a look at cryptography and generation of randomness. The third one, by S.A. Molchanov, covers three aspects of random media: homogenization theory, loca- lization properties and intermittency. Each of these chap- ters provides an introduction to and survey of its subject.

Published
2006

48. Ricci Curvature and Dimension for Diffusion Semigroups

Author

Dominique Bakry

Subjects
Curvature of Riemannian manifolds, Semigroup, Bounded function, Mathematical analysis, Ricci flow, Mathematics::Differential Geometry, Riemannian manifold, Curvature, Ricci curvature, Mathematics, Scalar curvature
Abstract

This paper is a brief survey of some properties of the “iterated squared gradient” associated to some diffusion semigroup. In the first part, we use this notion to give an “intrinsic” definition of the Ricci curvature and of the dimension of the semigroup: in the case of the heat semigroup on a Riemannian manifold, we recover the usual notions. In the second part, we describe some properties of diffusions with Ricci curvature bounded from below. In the third part, we show how to improve these properties in the case of diffusions with finite dimension. The fourth part is devoted to examples, and was worked out with the help of M.Emery.

Published
1990
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