Ergodicity of Markov Processes Via Nonstandard Analysis.

Format:

Book

Author/Creator:

Duanmu, Haosui.

Contributor:

Rosenthal, Jeffrey S.

Weiss, William.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.273

Language:

English

Subjects (All):

Markov processes.

Ergodic theory.

Nonstandard mathematical analysis.

Physical Description:

1 online resource (126 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"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"-- Provided by publisher.

Contents:

Cover

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.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

Other Format:

Print version: Duanmu, Haosui Ergodicity of Markov Processes Via Nonstandard Analysis

ISBN:

9781470468132

1470468131

OCLC:

1284944707