马氏过程遍历或衰减速度的研究在概率理论研究中起着重要的作用。本硕士论文对两类特殊的马氏过程的遍历或衰减性质作了进一步的研究。对于非凸非紧流形上扩散过程，与一般凸流形上的扩散过程相比，要判断其是否代数式遍历的难点在于连接流形上任意两点的测地线未必全在流形上，所以单纯想通过测地距离来构造控制泛函V 很难验证其压缩性，本论文意在解决此问题，并给出一般的判别条件，最后利用比较定理对其进行了简化。论文的第二部分讨论了当生灭链非常返时，它具有代数式衰减速度的一些判别条件。由于非常返生灭链的可配称测度不可和， 所以必须对遍历情况下代数式收敛的等价形式（Liggett-Stroock 定理）作相应的修改，并构造相对较好的新的控制泛函，得到生灭链代数式衰减的判别条件和必要条件。在第一章中， 我们简要介绍了不同马氏过程代数式收敛或衰减问题的研究背景和已有结果， 并且总结了本文的一些主要结果。在第二章中， 我们研究了在非凸非紧流形上扩散过程代数式收敛的一些判别条件，并给出了收敛阶的定量估计。在第三章中， 研究了连续时间非常返生灭链代数式衰减的一些判别条件。文中我们构造了两类不同的泛函V ，针对不同的泛函，给出了相应的判别条件， 但是遗憾的是未能对第二种给出相应的必要条件。

It is well known that estimation of ergodic or decay rate for Markov processesplays an important role on the research of the probability theory. In this thesis,we discuss the ergodic or decay property of two special types of Markov processes.Firstly, we consider L2- algebraic ergodicity for the di®usion processes on non-convexmanifold. Compared with these on convex manifold, the main obstacle is that theminimal geodesic between two points on the manifold may not be entirely containedin this manifold. So it is useless to construct the controlling functional V by thegeodesic distance directly. The main purpose is to overcome these problem andpresent more general results. In the second part of these thesis, we discuss theproblem of the decay rate of the transient birth-death process. As the symmetricmeasure is in¯nite for the transient process, we have to modify the Liggett-Stroocktheorem and construct relatively good functional and get the criteria in the end.In chapter 1, we brie°y introduce the background of algebraic convergence ordecay for di®erent models of Markov processes, and summarize the main results ofthis paper.In chapter 2, we consider the algebraic convergence rate in L2-sense for di®usionprocesses on non-convex manifold, and a series of su±cient and necessary conditionsfor these convergence are presented.In chapter 3, we modify the equivalent formula of algebraic decay for transientbirth-death processes ¯rstly, and construct di®erent types of controlling functionalV , and deduce the su±cient and necessary criteria for algebraic decay with respectto them.