马氏过程的稳定性在马氏过程理论的研究中具有至关重要的作用, 其主要内容包括唯一性和常返性, 各种遍历性(遍历性, 指数遍历性, 强遍历性) 等. 对于生灭跳过程及单生跳过程和单死跳过程, 显式判别准则大多已经得到. 本文对单生跳过程和单死跳过程进行推广, 主要研究m 程跳过程的稳定性性质. 文章共分为四个部分.
第一部分, 考虑到向上m 程跳过程的可跨跳性, 对于唯一性问题难以使用经典的方法解决, 本文将唯一性转化为常返性问题从而与常返性问题一并得到了充分必要的显式判别准则. 对于其积分型泛函的矩, 采用最小非负解理论和截断逼近技术自然过渡到矩的显式表达. 与此同时, 也给出了遍历性与强遍历性的充分必要判别准则. 同样地, 也得到了其积分型泛函的Laplace 变换以及其击中时指数阶矩的显式表达. 最后, 通过实例验证了所给出的方法的有效性.
第二部分, 对于向下m 程跳过程, 类似于向上的情形进行处理, 但其截断扩充过程的形式较向上的情形有较大的不同. 考虑到其具有无穷流出性, 可以预料到表达式会较向上情形更为复杂. 其相应的表达式均已在本文中得到. 除此之外, 还归纳得到了双死跳过程不变测度的显式表达并通过实例验证.
第三部分, 针对单死跳过程, 解决了其向上或向下积分型泛函的分布显式表达的问题. 在一定条件下, 对于向下的情形, 根据已知的单死跳过程积分型泛函高阶矩的结果, 结合分布由矩决定这一事实, 得到了向下积分型泛函Laplace 变换的显式表达. 在向上的情形, 将最小非负解理论及已得到的向下的结果结合起来, 得到了向上情形的对应结果.
最后一部分, 同样作为单生跳过程和单死跳过程的推广, 考虑带一般边界条件的单生跳过程及单死跳过程, 基于相应不等式组解的构造方法, 给出了其唯一性, 常返性, 各类遍历性(遍历性, 指数遍历性, 强遍历性) 的显式判别准则。

The stability of Markov processes is of paramount importance in the study of Markov processes, which mainly includes some aspects such as uniqueness and recurrence, as well as various forms of ergodicity(ordinary ergodicity, exponential ergodicity, strong ergodicity). For birth-death jump processes and both single birth and single death jump processes, most explicit stability criteria have already been established by many researchers. This thesis extends the study to jump processes with uniformly finite range, focusing on their stability properties. The thesis is divided into four parts.

In the first part, in view of  the leapability of the uniformly finite range upward jump processes, the uniqueness problem is difficult to be sovled by classical methods. We transform the uniqueness problem into the recurrence one and thereby obtain explicit necessary and sufficient criteria for both problems. For the moments of their integral functionals, we employ the theory of minimal non-negative solutions and augmented truncation approximation techniques to naturally derive explicit expressions for the moments. Simultaneously, we provide necessary and sufficient criteria for ergodicity and strong ergodicity. Similarly, we derive explicit expressions for the Laplace transforms of the integral-type functionals and the exponential moments associated with the first hitting times. Finally, we validate the effectiveness of our methods through illustrative examples.

In the second part, we address uniformly finite range downward jump processes in the way analogously to the upward case but with significant differences in the form of the augmented truncation processes. Given that they possess property of infinite-exit, the resulting expressions are expectedly more complex than those in the upward case. All corresponding expressions are derived within this thesis. Additionally, we obtain and validate the explicit expressions for the invariant measures of the double death jump processes through examples.

In the third part, we solve the problem of distributions of integral-type functionals for single death jump processes, both upward and downward. Under certain conditions, for the downward case, leveraging known results for the higher moments of the integral functionals of single death jump processes and the fact that distributions are determined by moments, we derive explicit expressions for the Laplace transforms of the downward integral-type functionals. For the upward case, we combine the minimal non-negative solution theory with the results obtained for the downward case to derive the corresponding expressions.

In the final part, as the generalization of single birth jump processes and single death jump processes, we consider single birth and single death jump processes with general boundary conditions. Using the construction method of the solution to the  corresponding inequality, we provide explicit criteria for uniqueness, recurrence, and various forms of ergodicity(ordinary ergodicity, exponential ergodicity, strong ergodicity).