Dynamics of continuous time Markov chains with applications

Abstract

This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes are abundant in applications, in particular within biology. We study the classification of states for general CTMCs on the non-negative integer lattices, by characterizing the set of absorbing states (similarly, trapping, escaping, positive irreducible components and quasi-irreducible components). For CTMCs on non-negative integers with polynomial transition rates, we provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, positive/null recurrence, implosivity, and existence and non-existence of passage times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained. Moreover, an identity for stationary measures is established and tail asymptotics for stationary distributions is given. A similar identity as well as asymptotics is derived for quasi-stationary distributions. Finally, we apply our results to stochastic reaction networks.