本学位论文主要研究离散时间马尔可夫链的各种泛函不等式及其稳定性问题.在第二章中, 我们应用概率方法并结合经典的Poincar´e 不等式, 得到了离散时间可逆马尔可夫链谱隙和谱半径之间的定量关系. 这具有基本的重要性. 通过一个更新公式, 我们把此结果推广到L∞ 空间.在第三章中, 我们采用概率方法并结合Cheeger 等周常数, 讨论了离散时间可逆马尔可夫链的对数Sobolev 不等式和Nash 不等式.在第四章中, 我们系统地研究了离散时间马尔可夫链的随机稳定性, 包括几何非常返性和代数非常返性. 通过建立恰当的drift 条件以及利用修正的首次回返时的相应阶矩, 我们研究了各种非常返性的判别准则. 作为主要结果的应用, 我们给出了半直线上的随机游动和非负整数集上的单生链各种非常返性的显式判别.在第五章中, 我们进一步讨论了连续时间马尔可夫过程的指数非常返性, 给出了基于广义生成元的drift 条件. 此外, 我们还研究了若干典型的例子, 包括单生过程, 扩散算子以及L´evy 型算子.

In this thesis, we study various functional inequalities and stability for discretetimeMarkov chains.In Chapter 2, by combining a probabilistic method with the classical Poincar´einequality, the quantitative relationship between spectral gap and spectral radiusis obtained for the reversible discrete-time Markov chain. This is fundamentallyimportance. Via a renewal formula, these results are extended to L∞-space.In Chapter 3, we study the log-Sobolev and Nash inequalities for the reversiblediscrete-time Markov chain by using a probabilistic approach and the generalizedCheeger’s method.In Chapter 4, we systematically study the stochastic stability for the discretetimeMarkov chain, including the geometric and algebraic transience. Criteria arepresented through establishing the drift conditions and using the modified momentsof the first return time. As applications, we give explicit criteria for the randomwalk on the half line and the skip-free chain on nonnegative integers.In Chapter 5, we further study the exponential transience for the continuoustimeMarkov process. Criteria are presented through developing the drift conditionsfor the extended generator. Moreover, some typical examples are investigated, includingthe single-birth process, the diffusion operator and the L´evy-type operator.