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Autore: | Benaim Michel |
Titolo: | Markov chains on metric spaces : a short course / / Michel Benaim, Tobias Hurth |
Pubblicazione: | Cham, Switzerland : , : Springer, , 2022 |
©2022 | |
Descrizione fisica: | 1 online resource (205 pages) |
Disciplina: | 519.233 |
Soggetto topico: | Markov processes |
Metric spaces | |
Processos de Markov | |
Espais mètrics | |
Soggetto genere / forma: | Llibres electrònics |
Persona (resp. second.): | HurthTobias |
Nota di bibliografia: | Includes bibliographical references (pages 187-190) and index. |
Nota di contenuto: | Intro -- Preface -- Contents -- Preliminaries -- 1 Markov Chains -- 1.1 Markov Kernels -- 1.2 Markov Chains -- 1.3 The Canonical Chain -- 1.4 Markov and Strong Markov Properties -- 1.5 Continuous Time: Markov Processes -- 2 Countable Markov Chains -- 2.1 Recurrence and Transience -- 2.1.1 Positive Recurrence -- 2.1.2 Null Recurrence -- 2.2 Subsets of Recurrent Sets -- 2.3 Recurrence and Lyapunov Functions -- 2.4 Aperiodic Chains -- 2.5 The Convergence Theorem -- 2.6 Application to Renewal Theory -- 2.6.1 Coupling of Renewal Processes -- 2.7 Convergence Rates for Positive Recurrent Chains -- Notes -- 3 Random Dynamical Systems -- 3.1 General Definitions -- 3.2 Representation of Markov Chains by RDS -- Notes -- 4 Invariant and Ergodic Probability Measures -- 4.1 Weak Convergence of Probability Measures -- 4.1.1 Tightness and Prohorov's Theorem -- A Tightness Criterion -- 4.2 Invariant Measures -- 4.2.1 Tightness Criteria for Empirical Occupation Measures -- 4.3 Excessive Measures -- 4.4 Ergodic Measures -- 4.5 Unique Ergodicity -- 4.5.1 Unique Ergodicity of Random Contractions -- 4.6 Classical Results from Ergodic Theory -- 4.6.1 Poincaré, Birkhoff, and Ergodic Decomposition Theorems -- 4.7 Application to Markov Chains -- 4.8 Continuous Time: Invariant Probabilities for Markov Processes -- Notes -- 5 Irreducibility -- 5.1 Resolvent and ξ-Irreducibility -- 5.2 The Accessible Set -- 5.2.1 Continuous Time: Accessibility -- 5.3 The Asymptotic Strong Feller Property -- 5.3.1 Strong Feller Implies Asymptotic Strong Feller -- 5.3.2 A Sufficient Condition for the Asymptotic Strong Feller Property -- 5.3.3 Unique Ergodicity of Asymptotic Strong Feller Chains -- Notes -- 6 Petite Sets and Doeblin Points -- 6.1 Petite Sets, Small Sets, Doeblin Points -- 6.1.1 Continuous Time: Doeblin Points for Markov Processes -- 6.2 Random Dynamical Systems. |
6.3 Random Switching Between Vector Fields -- 6.3.1 The Weak Bracket Condition -- 6.4 Piecewise Deterministic Markov Processes -- 6.4.1 Invariant Measures -- 6.4.2 The Strong Bracket Condition -- 6.5 Stochastic Differential Equations -- 6.5.1 Accessibility -- 6.5.2 Hörmander Conditions -- Notes -- 7 Harris and Positive Recurrence -- 7.1 Stability and Positive Recurrence -- 7.2 Harris Recurrence -- 7.2.1 Petite Sets and Harris Recurrence -- 7.3 Recurrence Criteria and Lyapunov Functions -- 7.4 Subsets of Recurrent Sets -- 7.5 Petite Sets and Positive Recurrence -- 7.6 Positive Recurrence for Feller Chains -- 7.6.1 Application to PDMPs -- 7.6.2 Application to SDEs -- 8 Harris Ergodic Theorem -- 8.1 Total Variation Distance -- 8.1.1 Coupling -- 8.2 Harris Convergence Theorems -- 8.2.1 Geometric Convergence -- Aperiodic Small Sets -- 8.2.2 Continuous Time: Exponential Convergence -- 8.2.3 Coupling, Splitting, and Polynomial Convergence -- 8.3 Convergence in Wasserstein Distance -- A Monotone Class and Martingales -- A.1 Monotone Class Theorem -- A.2 Conditional Expectation -- A.3 Martingales -- Bibliography -- List of Symbols -- List of Symbols -- Index. | |
Titolo autorizzato: | Markov Chains on Metric Spaces |
ISBN: | 3-031-11822-7 |
9783031118210 | |
Formato: | Materiale a stampa |
Livello bibliografico | Monografia |
Lingua di pubblicazione: | Inglese |
Record Nr.: | 9910632475603321 |
Lo trovi qui: | Univ. Federico II |
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