Analysis and geometry of Markov diffusion operators
 Author/Creator
 Bakry, D. (Dominique), author.
 Language
 English.
 Publication
 Cham ; New York : Springer, [2014]
 Physical description
 xx, 552 pages ; 25 cm.
 Series
 Grundlehren der mathematischen Wissenschaften ; 348.
Access
Available online

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Unknown
QA274.7 .B35 2014

Unknown
QA274.7 .B35 2014
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Contributors
 Contributor
 Gentil, Ivan, author.
 Ledoux, Michel, 1958 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 527545) and index.
 Contents

 Markov semigroups, basics and examples
 Markov semigroups
 Markov processes and associated semigroups
 Markov semigroups, invariant measures and kernels
 ChapmanKolmogorov equations
 Infinitesimal generators and Carré du Champ operators
 FokkerPlanck equations
 Symmetrie Markov semigroups
 Dirichlet forms and spectral decompositions
 Ergodicity
 Markov chains
 Stochastic differential equations and diffusion processes
 Diffusion semigroups and operators
 Ellipticity and hypoellipticity
 Domains
 Summary of hypotheses (Markov semigroup)
 Working with Markov semigroups
 Curvaturedimension condition
 Notes and references
 Model examples
 Euclidean heat semigroup
 Spherical heat semigroup
 Hyperbolic heat semigroup
 The heat semigroup on a halfline and the Bessel semigroup
 The heat semigroup on the circle and on a bounded interval
 SturmLiouville aemigroups on an interval
 Diffusion semigroups associated with orthogonal polynomials
 Notes and references
 Symmetric Markov diffusion operators
 Markov triples
 Second order differential operators on a manifold
 Heart of darkness
 Summary of hypotheses (Markov triple)
 Notes and references
 Three model functional inequalities
 Poincaré inequalities
 The example of the OrnsteinUhlenbeck semigroup
 Poincaré inequalities
 Tensorization of Poincaré inequalities
 The example of the exponential measure, and exponential integrability
 Poincaré inequalities on the real line
 The Lyapunov function method
 Local Poincaré inequalities
 Poincaré inequalities under a curvaturedimension condition
 BrascampLieb inequalities
 Further spectral inequalities
 Notes and references
 Logarithmic Sobolev inequalities
 Logarithmic Sobolev inequalities
 Entropy decay and hypercontractivity
 Integrability of eigenvectors
 Logarithmic Sobolev inequalities and exponential integrability
 Local logarithmic Sobolev inequalities
 Infinitedimensional Harnack inequalities
 Logarithmic Sobolev inequalities under a curvaturedimension condition
 Notes and references
 Sobolev inequalities
 Sobolev inequalities on the model spaces
 Sobolev and related inequalities
 Ultracontractivity and heat kernel bounds
 Ultracontractivity and compact embeddings
 Tensorization of Sobolev inequalities
 Sobolev inequalities and Lipschitz functions
 Local Sobolev inequalities
 Sobolev inequalities under a curvaturedimension condition
 Conformai invariance of Sobolev inequalities
 GagliardoNirenberg inequalities
 Fast diffusion equations and Sobolev inequalities
 Notes and references
 Notes and references
 Related functional, isoperimetric and transportation inequalities
 Generalized functional inequalities
 Inequalities between entropy and energy
 Offdiagonal heat kernel bounds
 Examples
 Beyond nash inequalities
 Weak poincare inequalities
 Further families of functional inequalities
 Summary for the model example ssa
 Notes and references
 Capacity and isoperimetrictype inequalities
 Capacity inequalities and coarea formulas
 Capacity and sobolev inequalities
 Capacity and poincare and logarithmic sobolev inequalities
 Capacity and further functional inequalities
 Gaussian isoperimetrictype inequalities under a curvature condition
 Harnack inequalities revisited
 From concentration to isoperimetry
 Notes and references
 Optimal transportation and functional inequalities
 Optimal transportation
 Transportation cost inequalities
 Transportation proofs of functional inequalities
 HamiltonJacobi equations
 Hypercontractivity of solutions of hamiltonjacobi equations
 Transportation cost and logarithmic sobolev inequalities
 Heat flow contraction in wasserstein space
 Curvature of metrie measure spaces
 Notes and references
 Appendices
 Semigroups of bounded operators on a banach space
 The hilleyosida theory
 Symmetrie operators
 Friedrichs extension of positive operators
 Spectral decompositions
 Essentially selfadjoint operators
 Compact and HilbertSchmidt operators
 Notes and references
 Elements of stochastic calculus
 Brownian motion and stochastic integrals
 The itö formula
 Stochastic differential equations
 Diffusion processes
 Notes and references
 Basic notions in differential and riemannian geometry
 Differentiable manifolds
 Some elementary euclidean geometry
 Basic notions in riemannian geometry
 Riemannian distance
 The riemannian T and T2 operators
 Curvaturedimension conditions
 Notes and references
 Afterword
 Chicken "Gaston Gerard"
 Notation and list of symbols
 Bibliography
 Index.
 Publisher's 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 Poincare, 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 nonspecialists. Simultaneously, it covers a wide range of results and techniques from the early developments in the mideighties 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.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2014
 Responsibility
 Dominique Bakry, Ivan Gentil, Michel Ledoux.
 Series
 Grundlehren der mathematischen Wissenschaften, 00727830 ; 348 A series of comprehensive studies in mathematics ; 348
 Available in another form
 (GyWOH)har135027706
 ISBN
 9783319002262 (hd.bd.)
 9783319002279 (online)
 3319002260 (cloth)
 3319002279 (eBook)