论文标题
连续时间马尔可夫链的结构分类
Structural classification of continuous time Markov chains with applications
论文作者
论文摘要
本文是由随机反应网络理论的示例激励的。随机反应网络的$ q $ -matrix可以从反应图派生，反应图是一个边缘标记的有向图，编码了不变空间上关联的连续时间马尔可夫链的跳跃向量$ \ mathbb {n}^d_0 $。一个开放的问题是如何将空间$ \ mathbb {n}^d_0 $分解为中性，捕获和逃避状态，以及开放式和封闭的沟通类，以及是否可以单独从反应图中完成。这种一般的连续时间马尔可夫连锁店可以理解为出生死亡过程的自然概括，并结合了多种不同的出生和死亡机制。我们表征了$ \ mathbb {n}^d_0 $由一般$ q $ -matrix施加的，在$ \ mathbb {n}^d_0 $中，就跳跃向量及其相应的过渡速率功能而言，具有$ \ mathbb {n}^d_0 $的值。因此，设置不限于随机反应网络。此外，我们定义了两个$ q $ amatrices的结构等效性，并为结构对等提供了足够的条件。示例在应用中很丰富。我们将结果应用于随机反应网络，生态学中的Lotka-Volterra模型，系统生物学中的Envz-OMPR系统以及一类扩展的分支过程，其中一个是出生死亡过程。
This paper is motivated by examples from stochastic reaction network theory. The $Q$-matrix of a stochastic reaction network can be derived from the reaction graph, an edge-labelled directed graph encoding the jump vectors of an associated continuous time Markov chain on the invariant space $\mathbb{N}^d_0$. An open question is how to decompose the space $\mathbb{N}^d_0$ into neutral, trapping, and escaping states, and open and closed communicating classes, and whether this can be done from the reaction graph alone. Such general continuous time Markov chains can be understood as natural generalizations of birth-death processes, incorporating multiple different birth and death mechanisms. We characterize the structure of $\mathbb{N}^d_0$ imposed by a general $Q$-matrix generating continuous time Markov chains with values in $\mathbb{N}^d_0$, in terms of the set of jump vectors and their corresponding transition rate functions. Thus the setting is not limited to stochastic reaction networks. Furthermore, we define structural equivalence of two $Q$-matrices, and provide sufficient conditions for structural equivalence. Examples are abundant in applications. We apply the results to stochastic reaction networks, a Lotka-Volterra model in ecology, the EnvZ-OmpR system in systems biology, and a class of extended branching processes, none of which are birth-death processes.
