Ergodic Theory Of Markov Process

In this paper, the ergodic theory of Markov processes is studied by it’s corresponding dynamical systems.In the first part of the paper we introduce the basic concepts and knowledge of dynamical systems and Markov processes. And then we link the theory of dynamical systems with the theory of Markov processes by a shift operator. We now can guess the main method used in this paper:Markov processes can be studied by dynamical systems which is more familiar to us.The main aim of the second chapter is to explore the relationship between Markov processes and dynamical systems, including the relationship of their ergodicity,weak mixing and strong mixing. In the beginning of this chapter, we first give a sufficient condition for the existence of an invariant measure of Markov process, ie a method to find invariant measure. Then we study the Markov process with invariant measure and its corresponding power system. We find that the Markov process with the invariant measure and its corresponding dynamical system have the same ergodicity, weak mixing and strong mixing. In addition, in order to prove a necessary and sufficient condition of ergodicity of dynamical systemes we also introduce the Birkhoff ergodic theorem.The third part of this paper is concerned with existence of an invariant measure, the recurrence of Markov process and trajectories’ denseness of Markov process.Then we come to the main conclusions of this chapter:(1) Recurrence:Let{Qn,n∈N}, be a Markovian semigroup with an invariant measure μ,Z(·)is the corresponding canonical process. If μ is ergodic. then for arbitrary T∈Σ:μ(T)>0, the process Z(·) is recurrent with respect to T.(2) Density of trajectories:Let{Z(n), n∈N} be a Markov process with transition probability π and P is the corresponding Markov operator. Assume that P admits an ergodic invariant measure μ.For arbitrary closed set D(?)S which satisfies that μ(D)>0and T(D)(?)D. if prob(Z(0)∈D)=1. then prob(cl{Z(0),Z(1),...}=D)=1.Tomasz Szarek, Lasota and Myjak have proved that:Let P be a nonexpansive Makov operator. Assume that P admits a unique invariant measure μ*. Set A*=supp μ and denote by {xn, n∈N} a Markov chain corresponding to P such that prob(x0∈A*)=1. Then prob(cl{Z(0),Z(1),...}=D)=1. The conclusion of this articale is a generalization of their conclusion. What’s more, our method are also different.Compressed Markov systems exported by the diagram, is the promotion of the common iterated function system. In the fourth chapter we promote the relationship between the support set of invariant measure and the attractor of the iterated function system. Then we get the relationship between the support set of invariant measure and the attractor of irreducible compression Markov system.