Title Analysis and Geometry of Markov Diffusion Operators / Dominique Bakry, Ivan Gentil, Michel Ledoux

Published Cham : Springer, 2014

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Description 1 online resource (xx, 552 pages) Series Grundlehren der mathematischen Wissenschaften ; 348 Grundlehren der mathematischen Wissenschaften ; 348. Contents 880-01 Introduction -- Part I Markov semigroups, basics and examples: 1. Markov semigroups -- 2. Model examples -- 3. General setting -- Part II Three model functional inequalities: 4. Poincar inequalities -- 5. Logarithmic Sobolev inequalities -- 6. Sobolev inequalities -- Part III Related functional, isoperimetric and transportation inequalities: 7. Generalized functional inequalities -- 8. Capacity and isoperimetry-type inequalities -- 9. Optimal transportation and functional inequalities -- Part IV Appendices: A. Semigroups of bounded operators on a Banach space -- B. Elements of stochastic calculus -- C. Some basic notions in differential and Riemannian geometry -- Notations and list of symbols 880-01/(S Machine generated contents note: 1. Markov Semigroups -- 1.1. Markov Processes and Associated Semigroups -- 1.2. Markov Semigroups, Invariant Measures and Kernels -- 1.3. Chapman-Kolmogorov Equations -- 1.4. Infinitesimal Generators and Carré du Champ Operators -- 1.5. Fokker-Planck Equations -- 1.6. Symmetric Markov Semigroups -- 1.7. Dirichlet Forms and Spectral Decompositions -- 1.8. Ergodicity -- 1.9. Markov Chains -- 1.10. Stochastic Differential Equations and Diffusion Processes -- 1.11. Diffusion Semigroups and Operators -- 1.12. Ellipticity and Hypo-ellipticity -- 1.13. Domains -- 1.14. Summary of Hypotheses (Markov Semigroup) -- 1.15. Working with Markov Semigroups -- 1.16. Curvature-Dimension Condition -- 1.17. Notes and References -- 2. Model Examples -- 2.1. Euclidean Heat Semigroup -- 2.2. Spherical Heat Semigroup -- 2.3. Hyperbolic Heat Semigroup -- 2.4. Heat Semigroup on a Half-Line and the Bessel Semigroup -- 2.5. Heat Semigroup on the Circle and on a Bounded Interval -- 2.6. Sturm-Liouville Semigroups on an Interval -- 2.7. Diffusion Semigroups Associated with Orthogonal Polynomials -- 2.8. Notes and References -- 3. Symmetric Markov Diffusion Operators -- 3.1. Markov Triples -- 3.2. Second Order Differential Operators on a Manifold -- 3.3. Heart of Darkness -- 3.4. Summary of Hypotheses (Markov Triple) -- 3.5. Notes and References -- 4. Poincaré Inequalities -- 4.1. Example of the Ornstein-Uhlenbeck Semigroup -- 4.2. Poincaré Inequalities -- 4.3. Tensorization of Poincaré Inequalities -- 4.4. Example of the Exponential Measure, and Exponential Integrability -- 4.5. Poincaré Inequalities on the Real Line -- 4.6. Lyapunov Function Method -- 4.7. Local Poincaré Inequalities -- 4.8. Poincaré Inequalities Under a Curvature-Dimension Condition -- 4.9. Brascamp-Lieb Inequalities -- 4.10. Further Spectral Inequalities -- 4.11. Notes and References -- 5. Logarithmic Sobolev Inequalities -- 5.1. Logarithmic Sobolev Inequalities -- 5.2. Entropy Decay and Hypercontractivity -- 5.3. Integrability of Eigenvectors -- 5.4. Logarithmic Sobolev Inequalities and Exponential Integrability -- 5.5. Local Logarithmic Sobolev Inequalities -- 5.6. Infinite-Dimensional Harnack Inequalities -- 5.7. Logarithmic Sobolev Inequalities Under a Curvature-Dimension Condition -- 5.8. Notes and References -- 6. Sobolev Inequalities -- 6.1. Sobolev Inequalities on the Model Spaces -- 6.2. Sobolev and Related Inequalities -- 6.3. Ultracontractivity and Heat Kernel Bounds -- 6.4. Ultracontractivity and Compact Embeddings -- 6.5. Tensorization of Sobolev Inequalities -- 6.6. Sobolev Inequalities and Lipschitz Functions -- 6.7. Local Sobolev Inequalities -- 6.8. Sobolev Inequalities Under a Curvature-Dimension Condition -- 6.9. Conformal Invariance of Sobolev Inequalities -- 6.10. Gagliardo-Nirenberg Inequalities -- 6.11. Fast Diffusion Equations and Sobolev Inequalities -- 6.12. Notes and References -- 7. Generalized Functional Inequalities -- 7.1. Inequalities Between Entropy and Energy -- 7.2. Off-diagonal Heat Kernel Bounds -- 7.3. Examples -- 7.4. Beyond Nash Inequalities -- 7.5. Weak Poincaré Inequalities -- 7.6. Further Families of Functional Inequalities -- 7.7. Summary for the Model Example [æ]α -- 7.8. Notes and References -- 8. Capacity and Isoperimetric-Type Inequalities -- 8.1. Capacity Inequalities and Co-area Formulas -- 8.2. Capacity and Sobolev Inequalities -- 8.3. Capacity and Poincaré and Logarithmic Sobolev Inequalities -- 8.4. Capacity and Further Functional Inequalities -- 8.5. Gaussian Isoperimetric-Type Inequalities Under a Curvature Condition -- 8.6. Harnack Inequalities Revisited -- 8.7. From Concentration to Isoperimetry -- 8.8. Notes and References -- 9. Optimal Transportation and Functional Inequalities -- 9.1. Optimal Transportation -- 9.2. Transportation Cost Inequalities -- 9.3. Transportation Proofs of Functional Inequalities -- 9.4. Hamilton-Jacobi Equations -- 9.5. Hypercontractivity of Solutions of Hamilton-Jacobi Equations . -- 9.6. Transportation Cost and Logarithmic Sobolev Inequalities -- 9.7. Heat Flow Contraction in Wasserstein Space -- 9.8. Curvature of Metric Measure Spaces -- 9.9. Notes and References -- Appendix A Semigroups of Bounded Operators on a Banach Space -- A.1. Hille-Yosida Theory -- A.2. Symmetric Operators -- A.3. Friedrichs Extension of Positive Operators -- A.4. Spectral Decompositions -- A.5. Essentially Self-adjoint Operators -- A.6. Compact and Hilbert-Schmidt Operators -- A.7. Notes and References -- Appendix B Elements of Stochastic Calculus -- B.1. Brownian Motion and Stochastic Integrals -- B.2. Itô Formula -- B.3. Stochastic Differential Equations -- B.4. Diffusion Processes -- B.5. Notes and References -- Appendix C Basic Notions in Differential and Riemannian Geometry -- C.1. Differentiable Manifolds -- C.2. Some Elementary Euclidean Geometry -- C.3. Basic Notions in Riemannian Geometry -- C.4. Riemannian Distance -- C.5. Riemannian Γ and Γ2 Operators -- C.6. Curvature-Dimension Conditions -- C.7. Notes and References -- Chicken "Gaston Gérard." Summary The present volume is an extensive monograph on the analytic and geometric aspects of Markov diffusion operators. It focuses on the geometric curvature properties of the underlying structure in order to study convergence to equilibrium, spectral bounds, functional inequalities such as Poincar, Sobolev or logarithmic Sobolev inequalities, and various bounds on solutions of evolution equations. At the same time, it covers a large class of evolution and partial differential equations. The book is intended to serve as an introduction to the subject and to be accessible for beginning and advanced scientists and non-specialists. Simultaneously, it covers a wide range of results and techniques from the early developments in the mid-eighties to the latest achievements. As such, students and researchers interested in the modern aspects of Markov diffusion operators and semigroups and their connections to analytic functional inequalities, probabilistic convergence to equilibrium and geometric curvature will find it especially useful. Selected chapters can also be used for advanced courses on the topic Analysis wiskunde mathematics differentiaalmeetkunde differential geometry partial differential equations analyse analysis waarschijnlijkheidstheorie probability theory stochastische processen stochastic processes functionaalanalyse functional analysis Mathematics (General) Wiskunde (algemeen) Bibliography Includes bibliographical references and index Notes Print version record Subject Markov operators. MATHEMATICS -- Applied. MATHEMATICS -- Probability & Statistics -- General. Markov operators Diffusionsprozess Markov-Prozess Halbgruppe Form Electronic book Author Gentil, Ivan, author Ledoux, Michel, 1958- author. ISBN 9783319002279 3319002279 3319002260 9783319002262

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