Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

Abstract

This paper provides a full classification of the dynamics for continuous-time Markov chains (CTMCs) on the nonnegative integers with polynomial transition rate functions and without arbitrary large backward jumps. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence versus transience, positive recurrence versus null recurrence, certain absorption, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions as well as existence and nonexistence of moments of hitting times are also obtained. Similar simple sufficient conditions for the aforementioned dynamics together with their opposite dynamics are established for CTMCs with unbounded forward jumps. 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, none of which are birth–death processes. The approach is based on a mixture of Lyapunov–Foster-type results, the classical semimartingale approach, and estimates of stationary measures.

Stochastic reaction networks within interacting compartments

“…The fact that F sim takes values in the space of finite tuples is equivalent to M C being finite for all time, which in turn is equivalent to the fact that M C is not explosive, regardless of the choice of rate constants in H K . This is a standard result in the theory of 1-d mass action stochastic reaction networks; see for instance [25].…”

“…The (positive) recurrence of this model is already completely classified [25]. We state this classification now as a lemma.…”

“…Note that H, the compartment network, is not explosive for any choice of rate constants (see e.g. [25]). So with probability one F sim undergoes only finitely many compartment transitions in finite time.…”

The asymptotic tails of limit distributions of continuous-time Markov chains

“…If a CTMC jumps unidirectionally (e.g., it is a pure birth or a pure death process), then all stationary measures, if such exist, are concentrated on absorbing states [48]. In contrast, an absorbed pure birth process has no QSDs.…”

“…We first provide the tail asymptotics for QSDs. We point out that parameter conditions for the existence and ergodicity of QSDs are given in [48]. Moreover, if R − > R + , we have the following:…”