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Ergodicity of Markov Processes via Nonstandard Analysis
About this Title
Haosui Duanmu, Jeffrey S. Rosenthal and William Weiss
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 273, Number 1342
ISBNs: 978-1-4704-5002-1 (print); 978-1-4704-6813-2 (online)
DOI: https://doi.org/10.1090/memo/1342
Published electronically: November 4, 2021
Keywords: Nonstandard analysis,
nonstandard measure theory,
discrete-time Markov processes,
continuous-time Markov processes
Table of Contents
Chapters
- 1. Introduction
- 2. Markov Processes and the Main Result
- 3. Preliminaries: Nonstandard Analysis
- 4. Internal Probability Theory
- 5. Measurability of Standard Part Map
- 6. Hyperfinite Representation of a Probability Space
- 7. General Hyperfinite Markov Processes
- 8. Hyperfinite Representation for Discrete-time Markov Processes
- 9. Hyperfinite Representation for Continuous-time Markov Processes
- 10. Markov Chain Ergodic Theorem
- 11. The Feller Condition
- 12. Push-down Results
- 13. Merging of Markov Processes
- 14. Miscellaneous Remarks
Abstract
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.- Robert M. Anderson, A nonstandard representation for Brownian motion and Itô integration, Bull. Amer. Math. Soc. 82 (1976), no. 1, 99–101. MR 405581, DOI 10.1090/S0002-9904-1976-13976-6
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