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

Part of: Physiological, cellular and medical topics Markov processes Ergodic theory

Published online by Cambridge University Press:  12 September 2022

Chuang Xu ,
Mads Christian Hansen  and
Carsten Wiuf
Chuang Xu*
Affiliation:
Technical University of Munich
Mads Christian Hansen*
Affiliation:
University of Copenhagen
Carsten Wiuf*
Affiliation:
University of Copenhagen
*
*Postal address: Faculty of Mathematics, Technical University of Munich, 85748 Garching bei München, Germany. Email address: Chuang.Xu@ma.tum.de
**Postal address: Department of Mathematical Sciences, University of Copenhagen, Copenhagen, 2100, Denmark.
**Postal address: Department of Mathematical Sciences, University of Copenhagen, Copenhagen, 2100, Denmark.
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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.

Keywords

Density-dependent continuous-time Markov chainsstochastic reaction networksexplosivityrecurrencetransiencecertain absorptionpositive and null recurrencestationary and quasi-stationary distributions

MSC classification

Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 37A25: Ergodicity, mixing, rates of mixing
Secondary: 92C42: Systems biology, networks 92C40: Biochemistry, molecular biology 92D25: Population dynamics (general)
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 321 - 355
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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