LEADER 05053nam 22006613 450 001 9910956322403321 005 20231110223341.0 010 $a9781470468132 010 $a1470468131 035 $a(MiAaPQ)EBC6822201 035 $a(Au-PeEL)EBL6822201 035 $a(CKB)20058043500041 035 $a(RPAM)22487735 035 $a(OCoLC)1284944707 035 $a(EXLCZ)9920058043500041 100 $a20211209d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aErgodicity of Markov Processes Via Nonstandard Analysis 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2021. 210 4$dİ2018. 215 $a1 online resource (126 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.273 311 08$aPrint version: Duanmu, Haosui Ergodicity of Markov Processes Via Nonstandard Analysis Providence : American Mathematical Society,c2021 9781470450021 320 $aIncludes bibliographical references. 327 $aCover -- Title page -- Chapter 1. Introduction -- 1.1. Chapter Outline -- Chapter 2. Markov Processes and the Main Result -- Chapter 3. Preliminaries: Nonstandard Analysis -- 3.1. The Hyperreals -- 3.2. Nonstandard Extensions of General Metric Spaces -- Chapter 4. Internal Probability Theory -- 4.1. Product Measures -- 4.2. Nonstandard Integration Theory -- Chapter 5. Measurability of Standard Part Map -- Chapter 6. Hyperfinite Representation of a Probability Space -- Chapter 7. General Hyperfinite Markov Processes -- Chapter 8. Hyperfinite Representation for Discrete-time Markov Processes -- 8.1. General properties of the transition probability -- 8.2. Hyperfinite Representation for Discrete-time Markov Processes -- Chapter 9. Hyperfinite Representation for Continuous-time Markov Processes -- 9.1. Construction of Hyperfinite State Space -- 9.2. Construction of Hyperfinite Markov Processess -- Chapter 10. Markov Chain Ergodic Theorem -- Chapter 11. The Feller Condition -- 11.1. Hyperfinite Representation under the Feller Condition -- 11.2. A Weaker Markov Chain Ergodic Theorem -- Chapter 12. Push-down Results -- 12.1. Construction of Standard Markov Processes -- 12.2. Push down of Weakly Stationary Distributions -- 12.3. Existence of Stationary Distributions -- Chapter 13. Merging of Markov Processes -- Chapter 14. Miscellaneous Remarks -- Acknowledgement -- Bibliography -- Back Cover. 330 $a"The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aMarkov processes 606 $aErgodic theory 606 $aNonstandard mathematical analysis 606 $aMathematical logic and foundations -- Nonstandard models -- Nonstandard models in mathematics$2msc 606 $aMeasure and integration -- Miscellaneous topics in measure theory -- Nonstandard measure theory$2msc 606 $aProbability theory and stochastic processes -- Markov processes -- Discrete-time Markov processes on general state spaces$2msc 606 $aProbability theory and stochastic processes -- Markov processes -- Continuous-time Markov processes on general state spaces$2msc 615 0$aMarkov processes. 615 0$aErgodic theory. 615 0$aNonstandard mathematical analysis. 615 7$aMathematical logic and foundations -- Nonstandard models -- Nonstandard models in mathematics. 615 7$aMeasure and integration -- Miscellaneous topics in measure theory -- Nonstandard measure theory. 615 7$aProbability theory and stochastic processes -- Markov processes -- Discrete-time Markov processes on general state spaces. 615 7$aProbability theory and stochastic processes -- Markov processes -- Continuous-time Markov processes on general state spaces. 676 $a519.2/33 686 $a03H05$a28E05$a60J05$a60J25$2msc 700 $aDuanmu$b Haosui$01801697 701 $aRosenthal$b Jeffrey S$0281907 701 $aWeiss$b William$01801698 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910956322403321 996 $aErgodicity of Markov Processes Via Nonstandard Analysis$94347070 997 $aUNINA