论文标题
一维连续时间马尔可夫链的动力学的完整分类，具有多项式过渡速率
Full classification of dynamics for one-dimensional continuous time Markov chains with polynomial transition rates
论文作者
论文摘要
本文为具有多项式过渡速率函数的非阴性整数上的连续时间马尔可夫链（CTMC）提供了全面分类。这种随机过程在应用中很丰富，尤其是在生物学中。更确切地说，对于有界跳跃的CTMC，我们在可计算的参数方面提供了必要和充分的条件，以实现爆炸性，复发性与瞬变，某些吸收，正面复发，无效复发和内爆。还可以获得简单的足够条件，以实现固定分布和准平台分布的指数性成分以及打击时间的存在和不存在。对于具有无限跳跃的CTMC，可以为上述动力学及其相反的动态提供类似的简单条件。结果概括了Karlin和McGregor在1960年代针对出生死亡过程的各个标准。最后，我们将结果应用于随机反应网络，一类扩展的分支过程，一般爆发的单细胞随机基因表达模型和人口过程，其中均不是出生死亡过程。该方法基于Lyapunov-Foster型结果，半明星方法的混合物以及固定度量的估计。
This paper provides full classification of dynamics for continuous time Markov chains (CTMCs) on the non-negative integers with polynomial transition rate functions. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence vs transience, certain absorption, positive recurrence vs null recurrence, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions as well as existence and non-existence of moments of hitting times are also obtained. Similar simple sufficient conditions for the aforementioned dynamics together with their opposite dynamics are established for CTMCs with unbounded jumps. The results generalize respective criteria for birth-death processes by Karlin and McGregor in the 1960s. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes, none of which are birth-death processes. The approach is based on a mixture of Lyapunov-Foster type results, semimartingale approach, as well as estimates of stationary measures.
