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 ${\mathbb{N}}_0^d$. An open question is how to decompose the space ${\mathbb{N}}_0^d$ 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 ${\mathbb{N}}_0^d$ imposed by a general Q-matrix generating continuous time Markov chains with values in ${\mathbb{N}}_0^d$, 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.

本文的灵感来自于随机反应网络理论中的例子。随机反应网络的Q矩阵可以从反应图导出，反应图是一个边标记的有向图，编码不变空间上相关连续时间马尔可夫链的跳跃向量${\mathbb{ñ}}_0^d$. 一个悬而未决的问题是如何分解空间${\mathbb{ñ}}_0^d$分为中立、诱捕和逃逸状态，以及开放和封闭的通信类，以及这是否可以仅从反应图中完成。这种一般的连续时间马尔可夫链可以理解为生死过程的自然概括，结合了多种不同的生死机制。我们描述了结构${\mathbb{ñ}}_0^d$由生成连续时间马尔可夫链的一般Q矩阵强加，其值为${\mathbb{ñ}}_0^d$，根据跳跃向量的集合及其相应的转换率函数。因此，设置不限于随机反应网络。此外，我们定义了两个Q-矩阵的结构等价，并为结构等价提供了充分条件。应用中的例子非常丰富。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，这些过程都不是生死过程。