13 results on '"Dominique Bakry"'
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2. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré
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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)
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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.
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- 2008
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3. A logarithmic Sobolev form of the Li-Yau parabolic inequality
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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)
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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.
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- 2006
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4. Some New Results on Eigenvectors via Dimension, Diameter, and Ricci Curvature
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Zhongmin Qian and Dominique Bakry
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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.
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- 2000
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5. On Harnack estimates for positive solutions of the heat equation on a complete manifold
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Dominique Bakry and Zhongmin Qian
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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.
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- 1997
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6. Optimal heat kernel bounds under logarithmic Sobolev inequalities
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Daniel Concordet, Michel Ledoux, and Dominique Bakry
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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.
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- 1997
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7. Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator
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Dominique Bakry and Michel Ledoux
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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.
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- 1996
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8. Weighted Nash Inequalities
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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)
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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
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- 2012
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9. On gradient bounds for the heat kernel on the Heisenberg group
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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
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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
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- 2007
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10. Perturbations of functional inequalities using growth conditions
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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)
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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.
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- 2007
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11. Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator
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Dominique Bakry and Michel Ledoux
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Pure mathematics ,Generator (computer programming) ,Markov chain ,General Mathematics ,Mathematical analysis ,Poincaré inequality ,53C20 ,58G30 ,Sobolev inequality ,symbols.namesake ,symbols ,47D07 ,Mathematics ,Sobolev spaces for planar domains - Published
- 1996
12. Ricci Curvature and Dimension for Diffusion Semigroups
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Dominique Bakry
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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.
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- 1990
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13. The Riesz transforms associated with second order differential operators
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Dominique Bakry
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Pure mathematics ,Riesz transform ,M. Riesz extension theorem ,Riesz potential ,Riesz representation theorem ,Singular integral operators of convolution type ,Mathematical analysis ,Operator theory ,Differential operator ,Fourier integral operator ,Mathematics - Abstract
This is an expository paper about some recent works on the Riesz transforms associated with a general symmetric, elliptic second order differential operator. Except in the third part, there are no new results and the proofs are often ommited or just sketched.
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- 1989
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