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Your search keyword '"Functional inequalities"' showing total 34 results
Start Over Descriptor "Functional inequalities" Topic 010102 general mathematics
34 results on '"Functional inequalities"'

1. Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators

Author

Menegaki, Angeliki, Menegaki, A [0000-0003-1192-3567], Apollo - University of Cambridge Repository, and Menegaki, Angeliki [0000-0003-1192-3567]

Subjects
4902 Mathematical Physics, Spectral gap, FOS: Physical sciences, 5103 Classical Physics, 01 natural sciences, Article, Mathematics - Spectral Theory, Hypocoercivity, Exponential convergence, 010104 statistics & probability, Mathematics - Analysis of PDEs, Chain (algebraic topology), FOS: Mathematics, 0101 mathematics, Spectral Theory (math.SP), Mathematical Physics, Physics, Steady state, Probability (math.PR), Nonequilibrium steady states, 4901 Applied Mathematics, 010102 general mathematics, Anharmonicity, Mathematical analysis, Functional inequalities, 4904 Pure Mathematics, Order (ring theory), Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Exponential function, Chain of oscillators, Heat bath, Hypoellipticity, Hypoelliptic operator, Bounded function, 49 Mathematical Sciences, 51 Physical Sciences, Mathematics - Probability, Analysis of PDEs (math.AP)
Abstract

We study a $1$-dimensional chain of $N$ weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with $N$ dependent bounds) perturbations of the harmonic ones. We show how a generalized version of Bakry-Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by F. Baudoin (2017). By that we prove exponential convergence to non-equilibrium steady state (NESS) in Wasserstein-Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than $N^{-3}$. For the purely harmonic chain the order of the rate is in $ [N^{-3},N^{-1}]$. This shows that, in this set up, the spectral gap decays at most polynomially with $N$., Comment: 54 pages, 1 figure, comments are welcome

Published
2020
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2. Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

Author

Eric A. Carlen and Jan Maas

Subjects
Computer Science::Machine Learning, Pure mathematics, Non-commutative optimal transport, Lindblad equation, 46L57, 34D05, 47C90, FOS: Physical sciences, Computer Science::Digital Libraries, 01 natural sciences, Article, Sobolev inequality, Statistics::Machine Learning, Gradient flow, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Operator Algebras (math.OA), Commutative property, Mathematical Physics, Ricci curvature, Mathematics, Semigroup, 010102 general mathematics, Functional inequalities, Mathematics - Operator Algebras, Statistical and Nonlinear Physics, Differential calculus, Mathematical Physics (math-ph), Functional Analysis (math.FA), Mathematics - Functional Analysis, Computer Science::Mathematical Software, Dissipative system, Spectral gap, 010307 mathematical physics
Abstract

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $C^*$-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein--Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates., An error in Theorem 10.6 has been corrected

Published
2019
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3. Functional version for Furuta parametric relative operator entropy

Author

Mustapha Raïssouli and Shigeru Furuichi

Subjects
47N10, Functional approach, Operator entropies, 010103 numerical & computational mathematics, 01 natural sciences, Operator inequality, 47A63, FOS: Mathematics, Discrete Mathematics and Combinatorics, Entropy (information theory), Applied mathematics, 0101 mathematics, Parametric statistics, Mathematics, Convex analysis, Operator inequalities, Research, Applied Mathematics, lcsh:Mathematics, 46N10, 010102 general mathematics, Functional inequalities, 52A41, lcsh:QA1-939, Functional Analysis (math.FA), Mathematics - Functional Analysis, 46A20, 39B62, 46N10, 46A20, 47A63, 47N10, 39B62, 52A41, Analysis
Abstract

Functional version for the so-called Furuta parametric relative operator entropy is here investigated. Some related functional inequalities are also discussed. The theoretical results obtained by our functional approach immediately imply those of operator versions in a simple, fast, and nice way.

Published
2018
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4. On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials

Author

Bertrand Lods, Amit Einav, and José A. Cañizo

Subjects
Conjecture, Entropy (statistical thermodynamics), Functional inequalities, Entropy, Boltzmann equation, Soft potentials, Applied Mathematics, 010102 general mathematics, 01 natural sciences, Exponential function, 010101 applied mathematics, symbols.namesake, Rate of convergence, Boltzmann constant, symbols, Applied mathematics, 0101 mathematics, Algebraic number, Linear boltzmann equation, Analysis, Mathematics
Abstract

In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad's angular cut-off assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cut-off case and conjecture what we believe to be the right rate of convergence in that case.

Published
2018
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5. Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities

Author

Matteo Muratori, Maria J. Esteban, Michael Loss, Jean Dolbeault, 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), School of Mathematics - Georgia Institute of Technology, Georgia Institute of Technology [Atlanta], Dipartimento di Matematica, 'Francesco Brioschi', Politecnico di Milano [Milan] (POLIMI), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), and ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013)

Subjects
Mathematics(all), fast diffusion equation, best constants, Mathematics::Analysis of PDEs, symmetry breaking, 01 natural sciences, Instability, Critical point (mathematics), Rényi entropy, symbols.namesake, Mathematics - Analysis of PDEs, Primary: 26D10, 35B06, 35J20Secondary: 49K30, 35J60, 35K55, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Symmetry breaking, Uniqueness, 0101 mathematics, Fisher information, symmetry, Mathematical physics, Mathematics, flows, semilinear elliptic equations, 010102 general mathematics, rigidity results, Functional inequalities, Symmetry in biology, uniqueness, General Medicine, carré du champ, interpolation, Caffarelli-Kohn-Nirenberg inequalities, 010101 applied mathematics, Emden-Fowler transformation, symbols, optimal functions, weights, Heuristics, Analysis of PDEs (math.AP)
Abstract

International audience; We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli-Kohn-Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition which determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden-Fowler transformation have to be modified.

Published
2017
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6. Delta-convexity with given weights

Author

Roman Ger

Subjects
Delta, Radon-Nikodym property (RNP), Pure mathematics, General Mathematics, lcsh:Mathematics, delta (s, t)-convexity, 010102 general mathematics, delta (s, t)-convexity, 010103 numerical & computational mathematics, General Medicine, Absolute continuity, lcsh:QA1-939, 01 natural sciences, Convexity, 26b25, 39b62, absolute continuity, 26a51, functional inequalities, 0101 mathematics, Jensen delta convexity, delta convexity, Mathematics
Abstract

Some differentiability results from the paper of D.Ş. Marinescu & M. Monea [7] on delta-convex mappings, obtained for real functions, are extended for mappings with values in a normed linear space. In this way, we are nearing the completion of studies established in papers [2], [5] and [7].

Published
2020

7. A proof of the Shepp-Olkin entropy monotonicity conjecture

Author

Erwan Hillion, Oliver Johnson, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), University of Bristol [Bristol], Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), School of Mathematics, University of Bristol, BS8 1TW Bristol, United Kingdom, affiliation inconnue, and Hillion, Erwan

Subjects
FOS: Computer and information sciences, Statistics and Probability, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Computer Science - Information Theory, Monotonic function, [MATH] Mathematics [math], 01 natural sciences, mixing coefficients, Combinatorics, 010104 statistics & probability, FOS: Mathematics, functional inequalities, Entropy (information theory), 0101 mathematics, [MATH]Mathematics [math], Mathematics, Conjecture, Information Theory (cs.IT), Probability (math.PR), 010102 general mathematics, Poisson binomial distribution, 94A17, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60E15, Poisson–binomial distribution, Statistics, Probability and Uncertainty, entropy, Mathematics - Probability
Abstract

Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to conjecture that this distribution should be 'more random' when each coin is fairer. Indeed, Shepp and Olkin conjectured that the Shannon entropy of this distribution is monotonically increasing in this case. We resolve this conjecture, by proving that this intuition is correct. Our proof uses a construction which was previously developed by the authors to prove a related conjecture of Shepp and Olkin concerning concavity of entropy. We discuss whether this result can be generalized to $q$-R\'{e}nyi and $q$-Tsallis entropies, for a range of values of $q$., Comment: 16 pages

Published
2019
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8. Non asymptotic variance bounds and deviation inequalities by optimal transport

Author

Kevin Tanguy

Subjects
Statistics and Probability, 60G70, Gaussian, 010102 general mathematics, Order statistic, superconcentration, Variance (accounting), 01 natural sciences, Exponential function, 010104 statistics & probability, Delta method, symbols.namesake, monotone rearrangement, order statistics, Simple (abstract algebra), Coulomb, symbols, Applied mathematics, functional inequalities, 0101 mathematics, Statistics, Probability and Uncertainty, 60E15, Real line, 26D10, Mathematics
Abstract

The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\mathbb{R} ^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.

Published
2019

9. Finiteness of entropy for granular media equations

Author

Julian Tugaut, Probabilités, statistique, physique mathématique (PSPM), 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), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Granular media equation, Entropy, 010102 general mathematics, Functional inequalities, Granular media, 01 natural sciences, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, McKean-Vlasov diffusion, Entropy (information theory), Statistical physics, Wasserstein distance, 0101 mathematics, Mathematics
Abstract

The current work deals with the granular media equation whose probabilistic interpretation is the McKean–Vlasov diffusion. It is well known that the Laplacian provides a regularization of the solution. Indeed, for any t > 0, the solution is absolutely continuous with respect to the Lebesgue measure. It has also been proved that all the moments are bounded for positive t. However, the finiteness of the entropy of the solution is a new result which will be presented here.

Published
2019

10. Entropy production in nonlinear recombination models

Author

Pietro Caputo, Alistair Sinclair, Caputo, Pietro, and Sinclair, Alistair

Subjects
Statistics and Probability, Kullback–Leibler divergence, Entropy, Analogy, FOS: Physical sciences, Population dynamic, 01 natural sciences, Boltzmann equation, 010104 statistics & probability, symbols.namesake, Quadratic equation, FOS: Mathematics, population dynamics, Mathematics - Combinatorics, functional inequalities, Statistical physics, 0101 mathematics, Mathematical Physics, Mathematics, Nonlinear equation, Entropy production, 010102 general mathematics, Probability (math.PR), Probability and statistics, Mathematical Physics (math-ph), Functional inequalitie, Nonlinear system, Boltzmann constant, symbols, nonlinear equations, Combinatorics (math.CO), entropy, Mathematics - Probability
Abstract

We study the convergence to equilibrium of a class of nonlinear recombination models. In analogy with Boltzmann's H theorem from kinetic theory, and in contrast with previous analysis of these models, convergence is measured in terms of relative entropy. The problem is formulated within a general framework that we refer to as Reversible Quadratic Systems. Our main result is a tight quantitative estimate for the entropy production functional. Along the way we establish some new entropy inequalities generalizing Shearer's and related inequalities., 27 pages

Published
2018

11. Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics

Author

Manh Hong Duong, Upanshu Sharma, Mark A. Peletier, André Schlichting, Agnes Lamacz, Center for Analysis, Scientific Computing & Appl., and Applied Analysis

Subjects
Kullback–Leibler divergence, large-deviation rate functionals, General Mathematics, General Physics and Astronomy, Markov process, 01 natural sciences, 010104 statistics & probability, symbols.namesake, Langevin equation, Mathematics - Analysis of PDEs, 0102 Applied Mathematics, FOS: Mathematics, State space, functional inequalities, Statistical physics, 0101 mathematics, Gibbs measure, Langevin dynamics, Mathematical Physics, Mathematics, effective dynamics for SDEs, Applied Mathematics, 010102 general mathematics, coarse-graining, Statistical and Nonlinear Physics, Mathematik, symbols, Granularity, Analysis of PDEs (math.AP), Curse of dimensionality, relative entropy techniques
Abstract

In molecular dynamics and sampling of high dimensional Gibb s measures coarse-graining is an important technique to reduce the dimensionality of the pro blem. We will study and quantify the coarse-graining error between the coarse-grained dynamic s and an effective dynamics. The effective dynamics is a Markov process on the coarse-grained state spa ce obtained by a closure procedure from the coarse-grained coefficients. We obtain error estimates b oth in relative entropy and Wasserstein distance, for both Langevin and overdamped Langevin dynami cs. The approach allows for vecto- rial coarse-graining maps. Hereby, the quality of the chose n coarse-graining is measured by certain functional inequalities encoding the scale separation of t he Gibbs measure. The method is based on error estimates between solutions of (kinetic) Fokker-Pla nck equations in terms of large-deviation rate functionals

Published
2018

12. Around the entropic Talagrand inequality

Author

Giovanni Conforti, Luigia Ripani, Institut für Mathematik [Potsdam], Universität Potsdam, 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), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Pure mathematics, Class (set theory), Inequality, Generalization, media_common.quotation_subject, 01 natural sciences, large deviations, semigroups, 010104 statistics & probability, symbols.namesake, Mathematics::Probability, FOS: Mathematics, functional inequalities, 0101 mathematics, media_common, Mathematics, Concentration of measure, Semigroup, Probability (math.PR), 010102 general mathematics, Connection (mathematics), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], optimal transport, symbols, Large deviations theory, Schrödinger's cat, Mathematics - Probability
Abstract

In this article, we study generalization of the classical Talagrand transport-entropy inequality in which the Wasserstein distance is replaced by the entropic transportation cost. This class of inequalities has been introduced in the recent work (Probab. Theory Related Fields 174 (2019) 1–47), in connection with the study of Schrödinger bridges. We provide several equivalent characterizations in terms of reverse hypercontractivity for the heat semigroup, contractivity of the Hamilton–Jacobi–Bellman semigroup and dimension-free concentration of measure. Properties such as tensorization and relations to other functional inequalities are also investigated. In particular, we show that the inequalities studied in this article are implied by a Logarithmic Sobolev inequality and imply Talagrand inequality.

Published
2018

13. Curvature and transport inequalities for Markov chains in discrete spaces

Author

Max Fathi, Yan Shu, Department of Mathematics [Berkeley], University of California [Berkeley], University of California-University of California, Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), University of California [Berkeley] (UC Berkeley), and University of California (UC)-University of California (UC)

Subjects
Statistics and Probability, Pure mathematics, Inequality, media_common.quotation_subject, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Curvature, 01 natural sciences, 010104 statistics & probability, Mathematics - Metric Geometry, FOS: Mathematics, functional inequalities, Mathematics::Metric Geometry, 0101 mathematics, Ricci curvature, Mathematics, media_common, Markov chains, Markov chain, Probability (math.PR), 010102 general mathematics, Metric Geometry (math.MG), Functional Analysis (math.FA), Exponential function, Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], optimal transport, curvature, discrete spaces, Mathematics::Differential Geometry, Mathematics - Probability
Abstract

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and \'Emery, the exponential curvature-dimension condition of Bauer \textit{et al.} and the coarse Ricci curvature of Ollivier. We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_1$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet-Meyer's theorem under the curvature-dimension condition CD$(\kappa,\infty)$ of Bakry and \'Emery., Comment: 26 pages. Comments are welcome

Published
2018
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14. Multiscale functional inequalities in probability: Concentration properties

Author

Antoine Gloria, Mitia Duerinckx, Département de Mathématique [Bruxelles] (ULB), Faculté des Sciences [Bruxelles] (ULB), Université libre de Bruxelles (ULB)-Université libre de Bruxelles (ULB), Quantitative methods for stochastic models in physics (MEPHYSTO), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université libre de Bruxelles (ULB), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Paul Painlevé - UMR 8524 (LPP), and Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe

Subjects
Multiscale, Statistics and Probability, 01 natural sciences, Sobolev inequality, Mixing (mathematics), 0103 physical sciences, FOS: Mathematics, Applied mathematics, Ergodic theory, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, Random field, Probability (math.PR), 010102 general mathematics, Functional inequalities, Probabilités, Covariance, Ambient space, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Elliptic operator, Nonlinear system, Mathématiques, 010307 mathematical physics, Concentration of measure, Mathematics - Probability
Abstract

In a companion article we have introduced a notion of multiscale functional inequalities for functions X(A) of an ergodic stationary random field A on the ambient space Rd. These inequalities are multiscale weighted versions of standard Poincare, covariance, and logarithmic Sobolev inequalities. They hold for all the examples of fields A arising in the modelling of heterogeneous materials in the applied sciences whereas their standard versions are much more restrictive. In this contribution we first investigate the link between multiscale functional inequalities and more standard decorrelation or mixing properties of random fields. Next, we show that multiscale functional inequalities imply fine concentration properties for nonlinear functions X(A). This constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto, and to the corresponding quantitative stochastic homogenization results., SCOPUS: ar.j, info:eu-repo/semantics/published

Published
2017
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15. Ground States in the Diffusion-Dominated Regime

Author

Franca Hoffmann, Edoardo Mainini, José A. Carrillo, Bruno Volzone, and Engineering & Physical Science Research Council (EPSRC)

Subjects
35K55, 35K65, 49K20, General Mathematics, Mathematics, Applied, 01 natural sciences, Article, Convexity, 0101 Pure Mathematics, CRITICAL MASS, Mathematics - Analysis of PDEs, 0102 Applied Mathematics, Critical mass, KELLER-SEGEL MODEL, FOS: Mathematics, REGULARITY, Uniform boundedness, Uniqueness, 0101 mathematics, Diffusion (business), 49K20, Energy functional, Mathematics, Science & Technology, Applied Mathematics, 010102 general mathematics, Mathematical analysis, ASYMPTOTICS, 35K65, AGGREGATION, FUNCTIONAL INEQUALITIES, CHEMOTAXIS, EXISTENCE, 010101 applied mathematics, Physical Sciences, 35K55, PRINCIPLE, Balanced flow, CELL-ADHESION, Analysis, Stationary state, Analysis of PDEs (math.AP)
Abstract

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document}C∞ inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.

Published
2017

16. Divergence-free positive symmetric tensors and fluid dynamics

Author

Denis Serre, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Stress-energy tensor, gas dynamics, homogenization, FOS: Physical sciences, fluid dynamics, Physics - Classical Physics, Kinetic energy, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, Inviscid flow, FOS: Mathematics, Fluid dynamics, Stress–energy tensor, functional inequalities, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Mathematical Physics, Mathematical physics, Conservation laws, Physics, Conservation law, Applied Mathematics, 010102 general mathematics, Classical Physics (physics.class-ph), A priori estimate, kinetic equations, Euler equations, 010101 applied mathematics, 35Q35, 35Q20, Euler's formula, symbols, Analysis, Analysis of PDEs (math.AP)
Abstract

We consider $d\times d$ tensors $A(x)$ that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of $(\det A)^{\frac1{d-1}}$. We apply them to models of compressible inviscid fluids: Euler equations, Euler--Fourier, relativistic Euler, Boltzman, BGK, etc... We deduce an {\em a priori} estimate for a new quantity, namely the space-time integral of $\rho^{\frac1n}p$, where $\rho$ is the mass density, $p$ the pressure and $n$ the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.

Published
2017

17. Interpolation inequalities on the sphere: linear vs. nonlinear flows

Author

Jean Dolbeault, Maria J. Esteban, Michael Loss, 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), School of Mathematics - Georgia Institute of Technology, Georgia Institute of Technology [Atlanta], ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), and ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013)

Subjects
Inequality, media_common.quotation_subject, CD(rho, spectral gap inequality, improved inequalities, 01 natural sciences, Upper and lower bounds, heat flow, Sobolev inequality, symbols.namesake, Rigidity (electromagnetism), Mathematics - Analysis of PDEs, FOS: Mathematics, functional inequalities, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], nonlinear diffusion, Uniqueness, 0101 mathematics, Mathematics, media_common, flows, 010102 general mathematics, Mathematical analysis, semilinear elliptic equations, rigidity results, uniqueness, General Medicine, optimal constants, 16. Peace & justice, Interpolation, 010101 applied mathematics, carré du champ method, Nonlinear system, Poincaré inequality, N) condition, Poincaré conjecture, Exponent, symbols, 58J35, 26D10, 35J60, Analysis of PDEs (math.AP)
Abstract

International audience; This paper is devoted to sharp interpolation inequalities on the sphere and their proof using flows. The method explains some rigidity results and proves uniqueness in related semilinear elliptic equations. Nonlinear flows allow to cover the interval of exponents ranging from Poincaré to Sobolev inequality, while an intriguing limitation (an upper bound on the exponent) appears in the carré du champ method based on the heat flow. We investigate this limitation, describe a counter-example for exponents which are above the bound, and obtain improvements below.

Published
2017

18. Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

Author

Matteo Muratori, Jean Dolbeault, Matteo Bonforte, Bruno Nazaret, Departamento de Matemáticas, Universidad Autonoma de Madrid (UAM), 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), Università degli Studi di Pavia, Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), and ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013)

Subjects
fast diffusion equation, best constants, linearization, 01 natural sciences, Power law, symmetry breaking, spectrum, Mathematics - Analysis of PDEs, 35K55, 46E35, 49K30, 26D10, 35B06, 49K20, 35J20, Linearization, Hardy-Poincaré inequality, Best constants, Caffarelli-Kohn-Nirenberg inequalities, Entropy methods, Fast diffusion equation, Flows, Functional inequalities, Interpolation, Optimal functions, Semilinear elliptic equations, Spectral gap, Spectrum, Symmetry, Symmetry breaking, Weights, Numerical Analysis, Modeling and Simulation, spectral gap, FOS: Mathematics, Entropy (information theory), functional inequalities, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, symmetry, entropy methods, flows, Entropy production, Numerical analysis, 010102 general mathematics, Mathematical analysis, semilinear elliptic equations, 010101 applied mathematics, Explicit symmetry breaking, optimal functions, weights, Analysis of PDEs (math.AP)
Abstract

International audience; In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not ? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy - entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN). We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy - entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincaré inequality which is interpreted as a linearized entropy - entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods. Consequences for the (WFD) flow will be studied in Part II of this work.

Published
2017

19. Entropic Ricci curvature bounds for discrete interacting systems

Author

Max Fathi, Jan Maas, Institute of Science and Technology [Klosterneuburg, Austria] (IST Austria), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Institute of Science and Technology [Austria] (IST Austria)

Subjects
Statistics and Probability, Work (thermodynamics), Pure mathematics, General method, 01 natural sciences, 010104 statistics & probability, FOS: Mathematics, functional inequalities, 0101 mathematics, [MATH]Mathematics [math], Discrete Ricci curvature, Ricci curvature, birth-death processes, Mathematics, Markov chain, Bernoulli–Laplace model, Probability (math.PR), 010102 general mathematics, Complete graph, random transposition model, Birth–death process, Functional Analysis (math.FA), Mathematics - Functional Analysis, 60K35, Product (mathematics), 60J10, Mathematics::Differential Geometry, Statistics, Probability and Uncertainty, transport metrics, zero-range processes, Mathematics - Probability
Abstract

We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli-Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition, we obtain new Ricci curvature bounds for zero-range processes on the complete graph. The method is inspired by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities., Comment: Published at http://dx.doi.org/10.1214/15-AAP1133 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2016
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20. Mass transportation proofs of free functional inequalities, and free Poincaré inequalities

Author

Ionel Popescu, 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), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Georgia Institute of Technology [Atlanta], 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
Mass transport, Pure mathematics, Spectral gap, Distribution (number theory), Context (language use), Mathematical proof, 01 natural sciences, 010104 statistics & probability, symbols.namesake, FOS: Mathematics, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Mathematics, Probability (math.PR), 010102 general mathematics, Functional inequalities, Free probability, Functional Analysis (math.FA), Mathematics - Functional Analysis, Poincaré conjecture, symbols, Random matrices, Convex function, Random matrix, Mathematics - Probability, Analysis
Abstract

This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincar\'e inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on $\R_{+}$ with the reference example of the Marcenko-Pastur distribution., Comment: This paper will apear in Journal of Fucntional Analysis

Published
2009
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21. Weighted interpolation inequalities: a perturbation approach

Author

Matteo Muratori, Bruno Nazaret, Jean Dolbeault, 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), Dipartimento di Matematica, 'Francesco Brioschi', Politecnico di Milano [Milan] (POLIMI), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), ANR-12-BS01-0019,STAB,Stabilité du comportement asymptotique d'EDP, de processus stochastiques et de leurs discrétisations.(2012), and ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013)

Subjects
Gamma-convergence, Inequality, General Mathematics, media_common.quotation_subject, Optimal functions, Mathematics::Analysis of PDEs, Perturbation (astronomy), 49J40, 46E35, 35B06, 26D10, 01 natural sciences, Symmetry, Concentration-compactness, Mathematics - Analysis of PDEs, Weights, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Best constants, Mathematics, media_common, 010102 general mathematics, Mathematical analysis, Symmetry in biology, Functional inequalities, 010101 applied mathematics, Symmetrization, Spectral gap, Perturbation method, Analysis of PDEs (math.AP)
Abstract

International audience; We study optimal functions in a family of Caffarelli-Kohn-Nirenberg inequalities with a power-law weight, in a regime for which standard symmetrization techniques fail. We establish the existence of optimal functions, study their properties and prove that they are radial when the power in the weight is small enough. Radial symmetry up to translations is true for the limiting case where the weight vanishes, a case which corresponds to a well-known subfamily of Gagliardo-Nirenberg inequalities. Our approach is based on a concentration-compactness analysis and on a perturbation method which uses a spectral gap inequality. As a consequence, we prove that optimal functions are explicit and given by Barenblatt-type profiles in the perturbative regime.

Published
2015
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22. On $\mathcal H^1$ and entropic convergence for contractive PDMP

Author

Pierre Monmarché, Monmarché, Pierre, Laboratoire Jacques-Louis Lions (LJLL), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Class (set theory), Markov process, 01 natural sciences, 010104 statistics & probability, symbols.namesake, Convergence (routing), PDMP, FOS: Mathematics, functional inequalities, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Invariant (mathematics), Mathematics, Discrete mathematics, Probability (math.PR), 010102 general mathematics, Variance (accounting), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Rate of convergence, Piecewise, symbols, Statistics, Probability and Uncertainty, 60J75, Mathematics - Probability
Abstract

Explicit rate of convergence in variance (or more general entropies) is obtained for a class of Piecewise Deterministic Markov Processes such as the TCP process, relying on functional inequalities. A method to establish Poincar\'e (and more generaly Beckner) inequalities with respect to a diffusion-type energy for the invariant law of such hybrid processes is devel- opped.

Published
2014

23. FROM LOG SOBOLEV TO TALAGRAND: A QUICK PROOF

Author

Michel Ledoux, Nicola Gigli, Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), 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), Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), 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, log-Sobolev inequality, Space (mathematics), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Measure (mathematics), Metric measure spaces, Sobolev inequality, 010104 statistics & probability, symbols.namesake, Development (topology), Settore MAT/05 - Analisi Matematica, Mathematics::Metric Geometry, Discrete Mathematics and Combinatorics, functional inequalities, 0101 mathematics, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG], Mathematics, Applied Mathematics, 010102 general mathematics, Hilbert space, optimal transport, Talagrand inequality, Sobolev space, Poincaré conjecture, Metric (mathematics), symbols, Analysis
Abstract

International audience; We provide yet another proof of the Otto-Villani theorem from the log Sobolev inequality to the Talagrand transportation cost inequality valid in arbitrary metric measure spaces. The argument relies on the recent development [2] identifying gradient flows in Hilbert space and in Wassertein space, emphasizing one key step as precisely the root of the Otto-Villani theorem. The approach does not require the doubling property or the validity of the local Poincare' inequality.

Published
2013

24. Lax type formulas with lower semicontinuous initial data and hypercontractivity results

Author

Paola Loreti, Antonio Avantaggiati, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome], Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate (MeMoMat), European Project: 264735,EC:FP7:PEOPLE,FP7-PEOPLE-2010-ITN,SADCO(2011), and Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome] (UNIROMA)

Subjects
Discrete mathematics, Class (set theory), Pure mathematics, functional inequalities, hamilton-jacobi equations, hopf-lax formulas, Logarithm, Applied Mathematics, 010102 general mathematics, Type (model theory), 16. Peace & justice, Lipschitz continuity, 01 natural sciences, Sobolev inequality, 010101 applied mathematics, Bounded function, Initial value problem, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Analysis, Mathematics
Abstract

International audience; In Avantaggiati and Loreti [Ric. Mat. 57(2):171-202, 2008] we studied the Cauchy problem for a class of Hamilton-Jacobi equations with initial data verifying the Lipschitz condition. In this paper we extend those results to the case in which the initial data are lower semicontinuous [in the following lsc], and are lower bounded and semiconvex. Here we prove hypercontractivity results and new Logarithm Sobolev Inequalities (shortly, LSI).

Published
2012
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25. Representation formula for the entropy and functional inequalities

Author

Joseph Lehec

Subjects
Statistics and Probability, Pure mathematics, Entropy, 010102 general mathematics, Probability (math.PR), Functional inequalities, Gaussian measure, 01 natural sciences, Functional Analysis (math.FA), Mathematics - Functional Analysis, 010104 statistics & probability, FOS: Mathematics, 39B62 (60H30), 39B62, 60J65, 0101 mathematics, Statistics, Probability and Uncertainty, Girsanov’s formula, Mathematics - Probability, Mathematics
Abstract

We prove a stochastic formula for the Gaussian relative entropy in the spirit of Borell's formula for the Laplace transform. As an application, we give unified and short proofs of a number of functional inequalities., Comment: 18 pages

Published
2010
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26. Intertwining and commutation relations for birth-death processes

Author

Aldéric Joulin, Djalil Chafaï, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), 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 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), ANR BLANC EVOL,ANR BLANC EVOL, Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), ANR-08-BLAN-0242,EVOL,Dissipative Evolutions and Convergence to Equilibrium(2008), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), 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 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
Statistics and Probability, Pure mathematics, Mathematics - Statistics Theory, Of the form, intertwining relation, Statistics Theory (math.ST), birth–death process, Curvature, 01 natural sciences, Convexity, 60J80, 39C05, 010104 statistics & probability, FOS: Mathematics, functional inequalities, Quantitative Biology::Populations and Evolution, Uniform boundedness, 0101 mathematics, Contraction (operator theory), Birth-death process, Mathematics, discrete gradients, Semigroup, 010102 general mathematics, Probability (math.PR), Feynman–Kac semigroup, Feynman-Kac semigroup, Lipschitz continuity, Birth–death process, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Probability
Abstract

Given a birth-death process on $\mathbb {N}$ with semigroup $(P_t)_{t\geq0}$ and a discrete gradient ${\partial}_u$ depending on a positive weight $u$, we establish intertwining relations of the form ${\partial}_uP_t=Q_t\,{\partial}_u$, where $(Q_t)_{t\geq0}$ is the Feynman-Kac semigroup with potential $V_u$ of another birth-death process. We provide applications when $V_u$ is nonnegative and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ433 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Published
2010
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27. Partitions and functional Santalo inequalities

Author

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

Subjects
TheoryofComputation_COMPUTATIONBYABSTRACTDEVICES, Inequality, General Mathematics, media_common.quotation_subject, MathematicsofComputing_NUMERICALANALYSIS, Computer Science::Computational Complexity, Computer Science::Computational Geometry, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Argument, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, FOS: Mathematics, Mathematics::Metric Geometry, Direct proof, 0101 mathematics, Mathematics, media_common, Computer Science::Cryptography and Security, Partition theorem, MSC: 39B62, 010102 general mathematics, Functional inequalities, TheoryofComputation_GENERAL, Inégalités fonctionnelles, Functional Analysis (math.FA), 010101 applied mathematics, Mathematics - Functional Analysis, TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES, 39B62, Mathematical economics, Logarithmic form
Abstract

We give a direct proof of a functional Santalo inequality due to Fradelizi and Meyer. This provides a new proof of the Blaschke-Santalo inequality. The argument combines a logarithmic form of the Prekopa-Leindler inequality and a partition theorem of Yao and Yao., Comment: 6 pages, file might be slightly different from the published version

Published
2010
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28. Stochastic Stokes' drift, homogenized functional inequalities, and large time behaviour of Brownian ratchets

Author

Jean Dolbeault, Michał Kowalczyk, Adrien Blanchet, Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille, 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), Depto Ingeniería Matemática (DIM), Facultad de Ciencias Fisicas y Matemáticas-Universidad de Santiago de Chile [Santiago] (USACH), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
intermediate asymptotics, Holley-Stroock perturbation results, Poincaré inequality, two-scale convergence, 01 natural sciences, functional inequalities, moment estimates, B- ECONOMIE ET FINANCE, Brownian motion, Mathematics, asymptotic expansion, logarithmic Sobolev inequalities, Applied Mathematics, Mathematical analysis, Fokker-Planck equation, traveling front, Gaussian measure, 010101 applied mathematics, Computational Mathematics, symbols, Fokker–Planck equation, 26D15, 35B27, 60E15, 80M40, loss of compactness, doubly-periodic equation, perturbation, homogenization, Homogenization (chemistry), Stochastic Stokes' drift, molecular motors, Sobolev inequality, symbols.namesake, traveling potential, spectral gap, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, sharp constants, Stokes drift, Brownian ratchets, defect of convergence, generalized Poincaré inequalities, 010102 general mathematics, minimizing sequences, contraction, Stokes flow, interpolation, transport, Analysis, effective diffusion
Abstract

International audience; A periodic perturbation of a Gaussian measure modifies the sharp constants in Poincaré and logarithmic Sobolev inequalities in the homogenization limit, that is, when the period of a periodic perturbation converges to zero. We use variational techniques to determine the homogenized constants and get optimal convergence rates towards equilibrium of the solutions of the perturbed diffusion equations. The study of these sharp constants is motivated by the study of the stochastic Stokes' drift. It also applies to Brownian ratchets and molecular motors in biology. We first establish a transport phenomenon. Asymptotically, the center of mass of the solution moves with a constant velocity, which is determined by a doubly periodic problem. In the reference frame attached to the center of mass, the behaviour of the solution is governed at large scale by a diffusion with a modified diffusion coefficient. Using the homogenized logarithmic Sobolev inequality, we prove that the solution converges in self-similar variables attached to the center of mass to a stationary solution of a Fokker-Planck equation modulated by a periodic perturbation with fast oscillations, with an explicit rate. We also give an asymptotic expansion of the traveling diffusion front corresponding to the stochastic Stokes' drift with given potential flow.

Published
2009

29. On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities

Author

Florent Malrieu, Djalil Chafaï, 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), 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 de Toulouse (UT)-Université de Toulouse (UT)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université de Toulouse (UT), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-08-BLAN-0242,EVOL,Dissipative Evolutions and Convergence to Equilibrium(2008), 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 ), Physiopathologie et Toxicologie Expérimentales ( UPTE ), Institut National de la Recherche Agronomique ( INRA ) -Ecole Nationale Vétérinaire de Toulouse, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), EVOL-ANR-09-Blanc,EVOL-ANR-09-Blanc, 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 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, AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), 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), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), and Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Pure mathematics, Poincare inequalities, Poincaré inequality, [ MATH.MATH-FA ] Mathematics [math]/Functional Analysis [math.FA], [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 60E15, 49Q20, 46E35, 62E99, 01 natural sciences, Sobolev inequality, 60E15, 49Q20, 46E35, 62E99, Tails probabilities, Gross logarithmic Sobolev inequalities, 010104 statistics & probability, symbols.namesake, Gaussian bounds, Mallows or Wasserstein distance, Probability theory, FOS: Mathematics, Mixtures of distributions, 0101 mathematics, Mathematics, Transportation cost distances, Laplace transform, Mass transportation, Concentration of measure, 010102 general mathematics, Probability (math.PR), Functional inequalities, Poincaré inequalities, Lipschitz continuity, Functional Analysis (math.FA), Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Sobolev space, Band isoperimetry, Transportation of measure, Bounded function, Deviation inequalities, symbols, Finite Gaussian mixtures, Statistics, Probability and Uncertainty, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability
Abstract

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincar\'e constant of a two-component mixture may remain bounded as the mixture proportion goes to 0 or 1 while the logarithmic Sobolev constant may surprisingly blow up. This counter-intuitive result is not reducible to support disconnections, and appears as a reminiscence of the variance-entropy comparison on the two-point space. As far as mixtures are concerned, the logarithmic Sobolev inequality is less stable than the Poincar\'e inequality and the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-component mixtures. This work leads to many open questions., Comment: Corrections. To appear in Annales de l'Institut Henri Poincare (AIHP)

Published
2008
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30. Functional inequalities and uniqueness of the Gibbs measure --- from log-Sobolev to Poincaré

Author

Pierre-André Zitt, Modélisation aléatoire de Paris X (MODAL'X), and Université Paris Nanterre (UPN)

Subjects
Statistics and Probability, Pure mathematics, unbounded spins, Finite volume method, 010102 general mathematics, MSC 2000: 82B20, 60K35, 26D10, Statistical mechanics, 01 natural sciences, Sobolev space, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, symbols.namesake, Convergence (routing), Ising model, symbols, functional inequalities, generalized Beckner inequalities, Uniqueness, Boundary value problem, 0101 mathematics, Gibbs measure, Heuristic argument, 82B20, 60K35, 26D10, Mathematics - Probability, Mathematics
Abstract

International audience; In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow the approach of G. Royer (1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box $[-n,n]^d$ (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sobolev, but stronger than Poincaré. We conclude by giving a heuristic argument showing that this could be the right inequalities to look at.

Published
2008
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31. 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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32. Interpolation between logarithmic Sobolev and Poincaré inequalities

Author

Jean-Philippe Bartier, Jean Dolbeault, Anton Arnold, Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster (WWU), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), 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), and Dolbeault, Jean

Subjects
Pure mathematics, Logarithm, perturbation, General Mathematics, convex Sobolev inequalities, MathematicsofComputing_NUMERICALANALYSIS, Poincaré inequality, 35K10, 01 natural sciences, AMS MSC 2000: 26D15, Sobolev inequality, symbols.namesake, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, spectral gap, functional inequalities, 46E35, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], logarithmic Sobolev inequality, hypercontractivity, Mathematics, 60J60, generalized Poincaré inequalities, logarithmic Sobolev inequalities, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Poincaré inequalities, interpolation, 010101 applied mathematics, Sobolev space, Poincaré conjecture, symbols, 39B62, Spectral gap, Logarithmic sobolev inequality, Interpolation, 60F10
Abstract

International audience; This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. Cette note est consacrée à des inégalités intermédiaires qui interpolent entre les inégalités de Sobolev logarithmiques et les inégalités de Poincaré. Pour de telles inégalités de Poincaré généralisées, nous améliorons les constantes données dans la littérature.

Published
2007

33. Binomial-Poisson entropic inequalities and the M/M/$\infty$ queue

Author

Djalil Chafaï, Laboratoire de Statistique et Probabilités (LSP), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), 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 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 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 Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National de la Recherche Agronomique (INRA), Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées

Subjects
Analyse fonctionnelle, Statistics and Probability, Poisson processes, M/M/1 queue, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Markov Processes, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Diffusion Processes, 010104 statistics & probability, Mathematics::Probability, Jump Processes, MSC 2000: 47D07, 60J60, 94A17, 26D15, 46E99, 60J27, 60J75, FOS: Mathematics, Poincare Inequalities, 0101 mathematics, Probability, Mathematics, Queueing theory, M/G/k queue, Classical Analysis and ODEs, 010102 general mathematics, Mathematical analysis, Probability (math.PR), Logarithmic Sobolev Inequalities, Probabilités, Heavy traffic approximation, M/M/∞ queue, Functional Analysis (math.FA), Mathematics - Functional Analysis, Functional Inequalities, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Analyse classique, Burke's theorem, 47D07, 60J60, 94A17, 26D15, 46E99, 60J27, 60J75, M/G/1 queue, M/M/c queue, Functional Analysis, Mathematics - Probability
Abstract

Accepted for publication in ESAIM/PS.; International audience; This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/$\infty$ queue. They describe in particular the exponential dissipation of $\Phi$-entropies along this process. This simple queueing process appears as a model of ``constant curvature'', and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/$\infty$ queues. Proofs are elementary and rely essentially on the development of a ``$\Phi$-calculus''.

Published
2005
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34. Semi Log-Concave Markov Diffusions

Author

Arnaud Guillin, Patrick Cattiaux, 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), Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-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
26D10, 35K55, 39B62, 47D07, 60J60, Markov chain, Semigroup, 010102 general mathematics, Mathematical analysis, Functional inequalities, diffusion processes, Coupling (probability), Curvature, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, convergence to equilibrium, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Simple (abstract algebra), Convergence (routing), 0101 mathematics, Diffusion (business), coupling, transport inequalities, Brownian motion, Mathematics
Abstract

In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under some ''curvature'' assumptions. Our notion of curvature coincides with the usual $\Gamma_2$ curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results for gradient/semigroup commutation in the log concave case. Some new convergence to equilibrium in the granular media equation is also exhibited.

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