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

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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.

Original languageEnglish
JournalAdvances in Applied Probability
Issue number1
Pages (from-to)321-355
Publication statusPublished - 2023

Bibliographical note

Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust.

    Research areas

  • certain absorption, Density-dependent continuous-Time Markov chains, explosivity, positive and null recurrence, recurrence, stationary and quasi-stationary distributions, stochastic reaction networks, transience

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