LEADER 04717nam 2200565 450 001 996499868703316 005 20240222151226.0 010 $a3-031-11822-7 010 $a9783031118210 035 $a(MiAaPQ)EBC7144578 035 $a(Au-PeEL)EBL7144578 035 $a(CKB)25456763000041 035 $a(PPN)266349269 035 $a(EXLCZ)9925456763000041 100 $a20230408d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aMarkov chains on metric spaces $ea short course /$fMichel Benaim, Tobias Hurth 210 1$aCham, Switzerland :$cSpringer,$d2022 210 4$d©2022 215 $a1 online resource (205 pages) 225 1 $aUniversitext 311 08$aPrint version: Benaïm, Michel Markov Chains on Metric Spaces Cham : Springer International Publishing AG,c2022 9783031118210 320 $aIncludes bibliographical references (pages 187-190) and index. 327 $aIntro -- 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 ?-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. 327 $a6.3 Random Switching Between Vector Fields -- 6.3.1 The Weak Bracket Condition -- 6.4 Piecewise Deterministic Markov Processes -- 6.4.1 Invariant Measures -- 6.4.2 The Strong Bracket Condition -- 6.5 Stochastic Differential Equations -- 6.5.1 Accessibility -- 6.5.2 Hörmander Conditions -- Notes -- 7 Harris and Positive Recurrence -- 7.1 Stability and Positive Recurrence -- 7.2 Harris Recurrence -- 7.2.1 Petite Sets and Harris Recurrence -- 7.3 Recurrence Criteria and Lyapunov Functions -- 7.4 Subsets of Recurrent Sets -- 7.5 Petite Sets and Positive Recurrence -- 7.6 Positive Recurrence for Feller Chains -- 7.6.1 Application to PDMPs -- 7.6.2 Application to SDEs -- 8 Harris Ergodic Theorem -- 8.1 Total Variation Distance -- 8.1.1 Coupling -- 8.2 Harris Convergence Theorems -- 8.2.1 Geometric Convergence -- Aperiodic Small Sets -- 8.2.2 Continuous Time: Exponential Convergence -- 8.2.3 Coupling, Splitting, and Polynomial Convergence -- 8.3 Convergence in Wasserstein Distance -- A Monotone Class and Martingales -- A.1 Monotone Class Theorem -- A.2 Conditional Expectation -- A.3 Martingales -- Bibliography -- List of Symbols -- List of Symbols -- Index. 410 0$aUniversitext. 606 $aMarkov processes 606 $aMetric spaces 606 $aProcessos de Markov$2thub 606 $aEspais mètrics$2thub 608 $aLlibres electrònics$2thub 615 0$aMarkov processes. 615 0$aMetric spaces. 615 7$aProcessos de Markov 615 7$aEspais mètrics 676 $a519.233 700 $aBenaim$b Michel$01267850 702 $aHurth$b Tobias 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 801 2$bUkBaUB 906 $aBOOK 912 $a996499868703316 996 $aMarkov Chains on Metric Spaces$92982350 997 $aUNISA