关于随机环境中的马尔可夫过程的简介——纪念李国平院士吴新谋教授诞辰100周年

摘要/Abstract

摘要：

该文系统地介绍随机环境中的马尔可夫过程. 共4章, 第一章介绍依时的随机环境中的马尔可夫链(MCTRE), 包括MCTRE的存在性及等价描述; 状态分类; 遍历理论及不变测度; p-θ 链的中心极限定理和不变原理. 第二章介绍依时的随机环境中的马尔可夫过程(MPTRE), 包括MPTRE的基本概念; 随机环境中的q -过程存在唯一性; 时齐的q -过程;MPTRE的构造及等价性定理.第三章介绍依时的随机环境中的分枝链(MBCRE), 包括有限维的和无穷维的MBCRE的模型和基本概念; 它们的灭绝概念;两极分化; 增殖率等.第四章介绍依时依空的随机环境中的马尔可夫链(MCSTRE), 包括MCSTRE的基本概念、构造; 依时依空的随机环境中的随机徘徊(RWSTRE)的中心极限定理、不变原理.

关键词: 依时随机环境中的马尔可夫链, 依时依空的随机环境中的马尔可夫链, 依时随机环境中的马尔可夫过程, 随机环境中的分枝链, 随机环境中的随机徘徊

Abstract:

We will introduce some results on Markov processes in random environments in this paper, there are four chapters in it. In chapter 1, we introduce the Markov chains in time random environments, including existence and equivalence; the chassification of states, ergodic theory and invariant measure, the central limit theorem and invariance principle for p-θ chain. In chapter 2, we introduce the Markov processes in time random environments, including some basic concepts, the exsistence and uniqueness for q-process in random environment, homogenious q-q-processes in random environments, the construction theorem and equivalence. In chapter 3 we introduce the branching chains in time random environments, including the models for finite dimension and infinite dimension, the extinction probability and palarization and
proliferattion rate are introduced. In chapter 4, we introduce the Markov chains in space time random environments, including basic concepts, central limit theorem and invariance priciple for RWSTRE.

Key words: Markov chains in time random environments, Markov chains in space time random environments, Markov processes in time random environments, Branching chains in random environments, Random walks in space time random environments

中图分类号:

60J05

胡迪鹤. 关于随机环境中的马尔可夫过程的简介——纪念李国平院士吴新谋教授诞辰100周年[J]. 数学物理学报, 2010, 30(5): 1210-1241.

HU Di-He. An Introduction to Markov Process in Random Environment[J]. Acta mathematica scientia,Series A, 2010, 30(5): 1210-1241.

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注:限于篇幅,略去参考文献近200篇,有兴趣的读者请参看作者近期要出版的专著---随机环境中的马尔可夫过程,其后面的参考文献有300多篇.