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Analysis and geometry of Markov diffusion operators / Dominique Bakry, Ivan Gentil, Michel Ledoux.

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Author/Creator:
Bakry, D. (Dominique), author.
Language:
English.
Publication date:
2014
Publication:
Cham ; New York : Springer, [2014]
Format:
  • Book
  • xx, 552 pages ; 25 cm.
Bibliography:
Includes bibliographical references (pages 527-545) and index.
Contents:
  • Markov semigroups, basics and examples
  • Markov semigroups
  • Markov processes and associated semigroups
  • Markov semigroups, invariant measures and kernels
  • Chapman-Kolmogorov equations
  • Infinitesimal generators and Carré du Champ operators
  • Fokker-Planck equations
  • Symmetrie Markov semigroups
  • Dirichlet forms and spectral decompositions
  • Ergodicity
  • Markov chains
  • Stochastic differential equations and diffusion processes
  • Diffusion semigroups and operators
  • Ellipticity and hypo-ellipticity
  • Domains
  • Summary of hypotheses (Markov semigroup)
  • Working with Markov semigroups
  • Curvature-dimension condition
  • Notes and references
  • Model examples
  • Euclidean heat semigroup
  • Spherical heat semigroup
  • Hyperbolic heat semigroup
  • The heat semigroup on a half-line and the Bessel semigroup
  • The heat semigroup on the circle and on a bounded interval
  • Sturm-Liouville 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 Ornstein-Uhlenbeck 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 curvature-dimension condition
  • Brascamp-Lieb 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
  • Infinite-dimensional Harnack inequalities
  • Logarithmic Sobolev inequalities under a curvature-dimension 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 curvature-dimension condition
  • Conformai invariance of Sobolev inequalities
  • Gagliardo-Nirenberg 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
  • Off-diagonal 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 isoperimetric-type inequalities
  • Capacity inequalities and co-area formulas
  • Capacity and sobolev inequalities
  • Capacity and poincare and logarithmic sobolev inequalities
  • Capacity and further functional inequalities
  • Gaussian isoperimetric-type 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
  • Hamilton-Jacobi equations
  • Hypercontractivity of solutions of hamilton-jacobi 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 hille-yosida theory
  • Symmetrie operators
  • Friedrichs extension of positive operators
  • Spectral decompositions
  • Essentially self-adjoint operators
  • Compact and Hilbert-Schmidt 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
  • Curvature-dimension conditions
  • Notes and references
  • Afterword
  • Chicken "Gaston Gerard"
  • Notation and list of symbols
  • Bibliography
  • Index.
Contributor:
Gentil, Ivan, author.
Ledoux, Michel, 1958- author.
Available in another form:
(GyWOH)har135027706
Series:
Grundlehren der mathematischen Wissenschaften, A series of comprehensive studies in mathematics ; 0072-7830 ; 348 348
Grundlehren der mathematischen Wissenschaften ; 348.
Subjects:
ISBN:
9783319002262
3319002260

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