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

Abstract

This paper studies the dynamics of continuous time Markov chains with possibly unbounded jump sizes on the non-negative integers with polynomial transition rate functions. Such stochastic processes are abundant in applications, in particular in biology. 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 identified and 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, none of which are birth-death processes.

Structural classification of continuous time Markov chains with applications

“…Hence they are structurally equivalent. Nonetheless, the associated CTMCs have quite different dynamics: The first network is explosive while the second is positive recurrent [2,18].…”

“…Nevertheless, the first SRN is is positive recurrent and admits an exponentially ergodic stationary distribution on N 0 . In contrast, the second reaction network is explosive a.s. for any initial state [18].…”

“…In addition, the classification result yields simple checkable criteria for irreducibility of one-dimensional CTMCs. Furthermore, it is found that even within SRNs with mass-action kinetics and d > 1, there are examples with denumerable irreducible components and very different dynamics on the individual components (for example, explosive vs. exponentially ergodic behavior) [18]. This phenomenon does not appear for d = 1.…”

“…Hence it is necessary to study the delicate structure of the ambient space, in order to fully understand the dynamics of the associated CTMCs. In a companion paper, the classification herein provides means to understand the dynamics of one-dimensional CTMCs [18].…”

Absolute concentration robustness in networks with low-dimensional stoichiometric subspace

“…Remark 2.5. One-dimensional networks have been analyzed recently in terms of multistationarity and multistability [26] and in the stochastic setting [28,29].…”