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This paper is motivated by demands in 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 an ambient space $mathbb{N}^d_0$. An open question is whether the ambient space $mathbb{N}^d_0$ can be decomposed into neutral, trapping, and escaping states, and open and closed communicating classes, based on the reaction graph alone. We characterize the structure of the state space of a $Q$-matrix on $mathbb{N}^d_0$ that generates continuous time Markov chains taking values in $mathbb{N}_0^d$, in terms of the set of jump vectors and their corresponding transition rate functions. We also define structural equivalence of two $Q$-matrices, and provide sufficient conditions for structural equivalence. Such stochastic processes are abundant in applications. We apply our 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.
This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices $N_0^d$, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes a
Continuous-time Markov chains are mathematical models that are used to describe the state-evolution of dynamical systems under stochastic uncertainty, and have found widespread applications in various fields. In order to make these models computation
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. Mo
We study certain properties of the function space of autocorrelation functions of Unit Continuous Time Markov Chains (CTMCs). It is shown that under particular conditions, the $L^p$ norm of the autocorrelation function of arbitrary finite state space
This paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions of continuous-time Markov chains on a subset of the non-negative integers. A new identity for stationary measures is established. In particular, f