非正则跳过程的随机稳定性

Markov processes is one of the major branches of probability theory. Its research content is very rich, and has been widely applied in science and practice. The jump process (that is continuous time Markov processes with denumerable state spaces) are mostly studied by assuming that the process corresponding to the generator matrix is unique (that is regular condition), for which rich and deep results have been derived. Comparing with this case, the study of the non-regular case is more difficult and has fewer existing references. However, by the construction theory of jump process, this project studies the stability of the minimal process for the non-regular case, such as the decay properties and its convergence rates etc., and the stability of other processes such as the ergodicity of the maximal process，which will be applied to many modern theory such as strong stationary time, cutoff phenomenon, quasi-stationary distribution and so on. Then the non-regular case will be clearer, which will play a certain role in the research of the non-regular jump process.

马氏过程是概率论研究的主要分支之一，研究内容极其丰富，且在其它学科和众多实际问题中都有广泛应用。目前，对于跳过程（即连续时间离散状态空间的马氏过程）的研究，大部分限定在生成矩阵对应过程唯一，即正则性条件下进行的，获得了众多而深刻的研究成果。相比之下，非正则跳过程的研究难度加大，相关文献也少。本课题借助跳过程的构造理论，主要研究非正则条件下最小跳过程的衰减性、收敛速度等稳定性质；同时，研究最小跳过程之上的其它过程的随机稳定性，如最大跳过程的遍历性等；并将结论应用于强平稳时、cutoff现象、拟平稳分布等国际前沿热点。这将使我们对非正则跳过程有较清晰的认识，对该过程的研究有一定的推动作用。

马氏过程的稳定性是概率论研究的经典和热点课题。对跳过程随机稳定性的研究，目前大都限定在正则条件下进行。相比之下，非正则跳过程的研究难度加大。本项目是我们对非正则跳过程随机稳定性的初步探索且取得较丰富的成果，主要包括正则边界可配称生灭过程的收敛速度和拟平稳分布，非正则单生过程的收敛速度和强平稳时分布，及一般非正则跳过程的遍历或衰减性及逼近。这些成果对非正则跳过程的研究具有很好的借鉴和推动作用，同时也为我们将来进行更加深入的研究奠定了坚实的基础。