Model of Markov chains in space-time random environments

Abstract

A general framework of stochastic model for a Markov chain in a space-time random environment is introduced, here the environment * , ,X is a random field. We study the dependence relations between the environment and the original chain, especially the "feedback". Some equivalence theorems and law of large numbers are obtained.The theory of stochastic processes in random environments has been pursued for some time, especially the random walk in random environment. Solomon [1] firstly gave the model of random walks in random environments. Kalikow [2] , Sinai [3] , and Sznaitman [4] studied deeply the model. At the same time, Nawrotzki [5] , and Cogburn [6][7][8] , also established a general model of Markov chains in random environments, and Orey [9] reviewed the works on this field and also gave some new results and open problems in a special invited paper. Please notice that the model of Solomon is space random environments, i.e. the environments only depend on the spatial locations, not evolution as time, and the model of Cogburn only considers the time but not the spatial locations of the original chain. Bérard [10] and Rassoul-Agha [11] studied a class of random walk in a space-time random environment in a sense, and obtained some results, such as central limit theorems and invariance principle. In the present paper, we establish a general formulation of Markov chains in space-time random environments, and give some equivalence theorems. Finally, we obtain a kind of law of large numbers of the model for general environments.

Cone Section Member Integration Design and Production Methods

“…The components that made by finite cone formed by the intersection of the cone [1] and other surface, widely used in space form because of the unique shape, if direct design and processing by its shape, it will influence application field because of the surface equation complicated or difficult processing technology . If we can according to the cone characteristics to use the plane plate processing, it will greatly simplify the production process to reduce cost.…”