Dynamics of continuous time Markov chains with applications

TL;DR

This paper provides a comprehensive analysis of continuous time Markov chains on non-negative integer lattices, focusing on polynomial transition rates, and applies these findings to various biological and stochastic models.

Contribution

It characterizes the structure of CTMCs on $ _0^d$, derives criteria for key dynamical properties, and applies results to biological and stochastic processes.

Findings

Criteria for explosivity, recurrence, and transience based on polynomial rates

Conditions for exponential ergodicity and quasi-stationary distributions

Applications to reaction networks, branching processes, and gene expression models

Abstract

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 are abundant in applications, in particular in biology. We characterize the structure of the state space of general CTMCs on $\N_0^d$ in terms of the set of jump vectors and their corresponding transition rate functions. For CTMCs on $\N_0$ with polynomial transition rate functions, we provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, certain absorption, positive/null recurrence, implosivity, and existence and non-existence of moments of hitting times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained, and the few gap cases are well-illustrated by examples. Subtle differences in conditions for different dynamical properties are revealed in terms of examples. 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 which are not birth-death processes.