Analysis and Geometry of Markov Diffusion Operators / Анализ и геометрия марковских диффузионных операторов
Год издания: 2014
Автор: Bakry D., Gentil I., Ledoux M. / Бакри Д., Жантиль И., Леду М.
Жанр или тематика: Монография
Издательство: Springer International Publishing
ISBN: 978-3-319-00227-9
Серия: Grundlehren der mathematischen Wissenschaften
Язык: Английский
Формат: PDF
Качество: Издательский макет или текст (eBook)
Интерактивное оглавление: Да
Количество страниц: 570
Тираж: нет данных
Описание:
Grundlehren der mathematischen Wissenschaften, vol. 348.
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 Poincaré, 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.
(перевод)
Настоящий том представляет собой обширную монографию по аналитическим и геометрическим аспектам марковских диффузионных операторов. Основное внимание уделяется геометрическим свойствам кривизны базовой структуры для изучения сходимости к равновесию, спектральным границам, функциональным неравенствам, таким как неравенства Пуанкаре, Соболева или логарифмические неравенства Соболева, а также различные оценки решений эволюционных уравнений. В то же время здесь охвачен и большой класс эволюционных и дифференциальных уравнений в частных производных.
Книга призвана служить введением в предмет и доступна как начинающим и продвинутым ученым, так и неспециалистам. Одновременно, она охватывает широкий спектр результатов и методов от ранних разработок середины восьмидесятых годов до последних достижений. Таким образом, студенты и исследователи, интересующиеся современными аспектами марковских диффузионных операторов и полугрупп и их связями с аналитическими функциональными неравенствами, вероятностной сходимостью к равновесию и геометрической кривизне, найдут её особенно полезной. Отдельные главы также могут быть использованы для продвинутых курсов по этой теме.
Оглавление
Preface vii
Basic Conventions xv
Contents xvii
Part I. Markov Semigroups, Basics and Examples
Chapter 1. Markov Semigroups 3
1.1 Markov Processes and Associated Semigroups 7
1.2 Markov Semigroups, Invariant Measures and Kernels 9
1.3 Chapman-Kolmogorov Equations 16
1.4 Infinitesimal Generators and Carré du Champ Operators 18
1.5 Fokker-Planck Equations 23
1.6 Symmetric Markov Semigroups 24
1.7 Dirichlet Forms and Spectral Decompositions 29
1.8 Ergodicity 32
1.9 Markov Chains 33
1.10 Stochastic Differential Equations and Diffusion Processes 38
1.11 Diffusion Semigroups and Operators 42
1.12 Ellipticity and Hypo-ellipticity 49
1.13 Domains 52
1.14 Summary of Hypotheses (Markov Semigroup) 53
1.15 Working with Markov Semigroups 56
1.16 Curvature-Dimension Condition 70
1.17 Notes and References 74
Chapter 2. Model Examples 77
2.1 Euclidean Heat Semigroup 78
2.2 Spherical Heat Semigroup 81
2.3 Hyperbolic Heat Semigroup 88
2.4 The Heat Semigroup on a Half-Line and the Bessel Semigroup 92
2.5 The Heat Semigroup on the Circle and on a Bounded Interval 96
2.6 Sturm-Liouville Semigroups on an Interval 97
2.7 Diffusion Semigroups Associated with Orthogonal Polynomials 102
2.8 Notes and References 118
Chapter 3. Symmetric Markov Diffusion Operators 119
3.1 Markov Triples 120
3.2 Second Order Differential Operators on a Manifold 137
3.3 Heart of Darkness 151
3.4 Summary of Hypotheses (Markov Triple) 168
3.5 Notes and References 173
Part II. Three Model Functional Inequalities
Chapter 4. Poincaré Inequalities 177
4.1 The Example of the Ornstein-Uhlenbeck Semigroup 178
4.2 Poincaré Inequalities 181
4.3 Tensorization of Poincaré Inequalities 185
4.4 The Example of the Exponential Measure, and Exponential Integrability 187
4.5 Poincaré Inequalities on the Real Line 193
4.6 The Lyapunov Function Method 201
4.7 Local Poincaré Inequalities 206
4.8 Poincaré Inequalities Under a Curvature-Dimension Condition 211
4.9 Brascamp-Lieb Inequalities 215
4.10 Further Spectral Inequalities 220
4.11 Notes and References 230
Chapter 5. Logarithmic Sobolev Inequalities 235
5.1 Logarithmic Sobolev Inequalities 236
5.2 Entropy Decay and Hypercontractivity 243
5.3 Integrability of Eigenvectors 250
5.4 Logarithmic Sobolev Inequalities and Exponential Integrability 252
5.5 Local Logarithmic Sobolev Inequalities 257
5.6 Infinite-Dimensional Harnack Inequalities 265
5.7 Logarithmic Sobolev Inequalities Under a Curvature-Dimension Condition 268
5.8 Notes and References 273
Chapter 6. Sobolev Inequalities 277
6.1 Sobolev Inequalities on the Model Spaces 278
6.2 Sobolev and Related Inequalities 279
6.3 Ultracontractivity and Heat Kernel Bounds 286
6.4 Ultracontractivity and Compact Embeddings 290
6.5 Tensorization of Sobolev Inequalities 291
6.6 Sobolev Inequalities and Lipschitz Functions 293
6.7 Local Sobolev Inequalities 296
6.8 Sobolev Inequalities Under a Curvature-Dimension Condition 305
6.9 Conformal Invariance of Sobolev Inequalities 313
6.10 Gagliardo-Nirenberg Inequalities 323
6.11 Fast Diffusion Equations and Sobolev Inequalities 330
6.12 Notes and References 340
Part III Related Functional, Isoperimetric and Transportation Inequalities
Chapter 7. Generalized Functional Inequalities 347
7.1 Inequalities Between Entropy and Energy 348
7.2 Off-diagonal Heat Kernel Bounds 355
7.3 Examples 362
7.4 Beyond Nash Inequalities 364
7.5 Weak Poincaré Inequalities 373
7.6 Further Families of Functional Inequalities 382
7.7 Summary for the Model Example μα 386
7.8 Notes and References 387
Chapter 8. Capacity and Isoperimetric-Type Inequalities 391
8.1 Capacity Inequalities and Co-area Formulas 392
8.2 Capacity and Sobolev Inequalities 396
8.3 Capacity and Poincaré and Logarithmic Sobolev Inequalities 399
8.4 Capacity and Further Functional Inequalities 403
8.5 Gaussian Isoperimetric-Type Inequalities Under a Curvature Condition 411
8.6 Harnack Inequalities Revisited 421
8.7 From Concentration to Isoperimetry 425
8.8 Notes and References 429
Chapter 9. Optimal Transportation and Functional Inequalities 433
9.1 Optimal Transportation 434
9.2 Transportation Cost Inequalities 438
9.3 Transportation Proofs of Functional Inequalities 442
9.4 Hamilton-Jacobi Equations 451
9.5 Hypercontractivity of Solutions of Hamilton-Jacobi Equations 454
9.6 Transportation Cost and Logarithmic Sobolev Inequalities 458
9.7 Heat Flow Contraction in Wasserstein Space 462
9.8 Curvature of Metric Measure Spaces 464
9.9 Notes and References 466
Appendices 471
Appendix A. Semigroups of Bounded Operators on a Banach Space 473
A.1 The Hille-Yosida Theory 473
A.2 Symmetric Operators 475
A.3 Friedrichs Extension of Positive Operators 477
A.4 Spectral Decompositions 478
A.5 Essentially Self-adjoint Operators 481
A.6 Compact and Hilbert-Schmidt Operators 483
A.7 Notes and References 485
Appendix B. Elements of Stochastic Calculus 487
B.1 Brownian Motion and Stochastic Integrals 487
B.2 The Itô Formula 491
B.3 Stochastic Differential Equations 493
B.4 Diffusion Processes 495
B.5 Notes and References 498
Appendix C. Basic Notions in Differential and Riemannian Geometry 499
C.1 Differentiable Manifolds 500
C.2 Some Elementary Euclidean Geometry 502
C.3 Basic Notions in Riemannian Geometry 504
C.4 Riemannian Distance 509
C.5 The Riemannian Г and Г2 Operators 511
C.6 Curvature-Dimension Conditions 513
C.7 Notes and References 518
Afterword 521
Chicken “Gaston Gérard” 521
Notation and List of Symbols 523
Bibliography 527
Index 547
