Markov Chain In Random Environment

As we all know, Markov chain is one of the most important stochastic processes, which is used in industry, agriculture, economy, insurance, biology, medicine and social science, etc.The transition matrix that is also called the environment of Markov chain is a key factor of Markov chain. Just as the nature environment is stochastic, we also make the environment of Markov chain be stochastic, then get the Markov Chain in Random Environment that is called MCRE for short. MCRE is divided in MCRE on time(the environment depend on time), MCRE on space(the environment depend on space) and MCRE on space and time(the environment depend on space and time together). This paper research MCRE on time only. For MCRE, many results are the generalization for corresponding results in classical Markov chains, many of others need to do much research. This paper summarizes the research job of predecessors, full text is divided in four part:The first part introduces the develop history and present conditions of MCRE;In the second part, the concept of average Markov process is introduced. Moreover we investigate the relations for Markov processes, martingales, stationary processes and average Markov process systematically.The third part summarizes the model that constructs the MCRE by p-m chains and skew product Markov chains by p-m chains, also we discuss the relations among the basic gunctions of the probability character for MCREIn the fourth part, we discuss the dimension of GW-tree and the canonical branching chain in random environment, furthermore we summarize the exact formulas of mathematical expectation and variance of branching chain in random environment.