Investigation On Some Topics About Ergodicity For Markov Processes

The purpose of this thesis is to investigate on some topics about ergodicity for Markov processes. It consists of the following three parts:estimates for the convergence rate of ergodic Markov chains, criteria for the sub-exponential ergodicity of Markov processes, and estimates for the princile eignavlue of symmeyric Levy type processes. To this end, we adopt the coupling method, Foster Lyapunov inequalities, and the approximation by q-processes. The thesis is divided into five chapters as follows.In Chapter One, we give some background about geometrically ergodic and subgeometrically ergodic Markov processes, and focus on applications for ergodic property of strong Markov processes. At the end, we also briefly recall some topics of our thesis.In Chapter Two, using the martingale technique and the coupling method, we obtain some bounds about the convergence rates for geometrically ergodic and subgeometrically ergodic Markov chains. For reversible case, we also establish the relation between (sub)geometrical ergodicity and (weak) Poincare inequality. Some examples for random walks on Z+are presented to illustrate to power of our results.In Chapter Three, we obtain the subexponential ergodicity for time-continuous Markov Processes via Foster Lyapunov inequalities. A number of examples are given to show the efficiency of our results. The relations with functional inequalities are also studied.In Chapter Four, we print out that the Cheeger argument does not work for the principle eigenvalue of symmetric Levy type operators, and also show that the corresponding eigenfunction is not monotone in general setting. Finally, using the approximation by symmetric q-processes, we obtain some estimates for the principle eigenvalue of symmetric Levy type operators.In Chapter Five, we summary main results of our thesis, and also present some questions for future research work.