Search

Your search keyword '"Dominique Bakry"' showing total 16 results
Start Over Author "Dominique Bakry" Topic 0101 mathematics
16 results on '"Dominique Bakry"'

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
Full Text
View/download PDF

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. 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
Full Text
View/download PDF

4. 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
Full Text
View/download PDF

5. 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
Full Text
View/download PDF

6. 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
Full Text
View/download PDF

7. 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
Full Text
View/download PDF

8. 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
Full Text
View/download PDF

9. 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
Full Text
View/download PDF

10. 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
Full Text
View/download PDF

11. 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
Full Text
View/download PDF

12. 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
Full Text
View/download PDF

13. 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
Full Text
View/download PDF

14. 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
Full Text
View/download PDF

15. 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
Full Text
View/download PDF

16. 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
Full Text
View/download PDF

Catalog

Books, media, physical & digital resources
See catalog results