The Existence And Uniqueness Of Markov-Feller Operators In Polish Spaces

The Markov-Feller operators, which originated in the study of ergodic properties of discrete-time homogeneous Markov chains, have appeared in the study of Feller processes, a type of Markov processes. The ergodic theory of these operators has been used extensively in many fields, for example, in dynamical systems, in the study of iterated function systems with probabilities, in the stability of solutions to stochastic differential equations, and in the study of convolutions of measures and so on.The existence of invariant measures, unique ergodicity and asymptotic stability are the main researching subjects of ergodic theory. They are also the main contents of the study of Markov-Feller operators. The unique ergodicity is a property which is weaker than asymptotic stability, but stronger than the existence of invariant probabilities. The asymptotic stability is a very strong property of Markov-Feller operators. There is a large class of Markov-Feller operators which has a unique invariant measure, but is not asymptotically stable. Therefore it is also a very important issue to characterize the unique ergodicity of Markov-Feller operators.We focus mainly on the existence and uniqueness of invariant probabilities of Markov-Feller operators in this paper. And the main method is to use the tightness of measure sequence and equicontinuity to ensure the existence of invariant measures and uniqueness. The main contents are as follows.The first chapter is introduction, which contains background and significance of this article, literature review and the main results.In chapter two we deal with the existence of invariant probabilities of Markov- Feller operators which are defined on Polish spaces. We assume that the Markov-Feller operators are generated by transition probability functions. The existence of invariant measures is always the most important issue in the study of ergodic theory of Markov operators. Krylov and Bogolioubov showed that a continuous transformation of a compact metric space must have an invariant measure. Lasta and Yorke gave sufficient conditions of existence of invariant measures and asymptotic stability of Markov operators in locally compact separate metric spaces using the asymptotic property of transition probability functions. S.Meyn and R.Tweedie proved that Foster-Lyapunov condition guarantees the existence of invariant probabilities of Markov operators in locally compact metric spaces. This condition is easy to use. In Polish spaces it is mainly through the tightness of iterated measure sequence to find the condition of existence of invariant probabilities. T.Szarek investigated the existence of invariant probabilities and asymptotic stability in Polish spaces. Various conditions are given to guarantee the existence of invariant probabilities of Markov operators. In this chapter, we give a sufficient condition, which states that if the dual operator of a Markov-Feller operator is equicontinuous in some point, the Markov-Feller operator has an invariant probability. Moreover, we weaken this condition and get another sufficient condition. The condition given in this chapter is easier to verify than other conditions.The unique ergodicity says that there is only one invariant measure of Markov operators. It is of much significance to characterize the unique ergodicity. Our goal in this chapter is to prove a criterion for the uniqueness of an invariant measure. Radu Zaharopol proved that the sufficient and necessary condition which guarantees the unique ergodicity of a Markov-Feller operator is that there exist dominant generic points defined by the operator. And he also gave a sufficient condition to ensure the unique ergodicity of a Markov-Feller operator utilizing the equicontinuity under the assumption that there is at least one invariant measure. T.Szarek gave a sufficient condition to ensure that there is at most one invariant measure of a Markov-Feller operator defined on a Polish space, which is showed that a Markov-Feller operator with equicontinuous dual operator which overlaps supports has at most one invariant measure. In chapter three, we continue the research work in Polish spaces, discussing the relationship between the unique ergodicity and the overlap of supports. We show that a Markov-Feller operator with equicontinuous dual operator is unique ergodic if and only if it overlaps supports.Finally, we study the ergodic theorem in Polish spaces in chapter four. P.Walters investigated continuous transformations in compact spaces and gave three conditions equivalent with unique ergodicity. We tried to extend it to Polish spaces. But studies have shown that these three conditions are not equivalent with unique ergodicity in Polish spaces. Therefore we just give a sufficient condition and state that the condition is not necessary by an example. Also, we discuss the ergodic measure in real number space R and summarize some application results of ergodic theory of Markov-Feller operators in Polish spaces.