Existence Of Invariant Measures And Unique Ergodicity For The Markov-Feller Operators Defined By Transition Probabilities

The Markov-Feller operators, which originate from the study of ergodicity of discrete-time homogeneous Markov chains, have appeared in the study of Feller processes, a class of Markov processes. The ergodic theory of these operators has been used extensively in many fields, such as dynamical systems, the study of iterated function systems with probabilities, the stability of solutions to stochastic differential equations, and the study of convolutions of measures etc. The existence of invariant probabilities and unique ergodicity are the main research subjects of ergodic theory and also the main contents of the study of Markov-Feller operators.Iterated Function Systems (IFS) originate from dynamical system. Dynamical theory deals with iteration of one map while IFS theory deals with iteration of many maps. At present, IFS is one of the important tools of studies and applications of fractal theory, autoregressive time series, image processing theory and stochastic dynamical system. And what’s more, the ideas of IFS exert a great influence on various fields including the theory of complex dynamical systems. The researches and extensions of IFS theory are of important theoretical values. The theory of finite IFS consisting of contraction mappings has been studied very thoroughly.The main results are as follows:The third chapter probes into the existence of invariant probabilities and unique ergodicity for Markov-Feller operators defined by transition probabilities on complete separate metric spaces, with method of using the tightness of measure sequences. The existence of invariant measures is always the most important issue in the study of ergodic theory of Markov operators. The unique ergodicity refers to that there is only one invariant measure of Markov operators which is stronger than the existence of invariant probabilities. Ergodic decomposition theorem states that ergodic measures are ’basic elements’ of invariant probabilities space and every invariant measure can be expressed by the integral of elementary ergodic measures. Ergodic decomposition theorem is of much significance to characterize the unique ergodicity. Yosida[30] provides ergodic decomposition theorem of Markov operators with invariant probabilities on compact spaces. Radu Zaharopol[25] studies the same problem on locally compact separate metric spaces and generalizes the conclusion of Yosida[30]. Moreover, the supports of ergodic measures are given. In this chapter, we provide the ergodic decomposition theorem for Markov-Feller operators generated by transition probability functions on complete separate metric spaces, and provide conditions to guarantee the existence of invariant probabilities and uniqueness of Markov-Feller operators.The fourth chapter probes into the ergodicity for Markov-Feller operators defined by infinite IFS on complete separate metric spaces. During the major theorems’proof procedure, we provide conditions to guarantee the existence of invariant probabilities and uniqueness for Markov-Feller operators. As a special class of Markov chains, the Markov-Feller operators defined by infinite IFS can be studied using Markov-Feller operator theory. This chapter adopts the method of analysis to explore the ergodicity for Markov-Feller operators defined by infinite IFS on Polish spaces. Theorem4.3.2makes a generalization from finite IFS on locally compact spaces to infinite IFS on complete separate metric spaces, and the methods of proof are much more elementary and simple.