TY - JOUR

T1 - Full classification of dynamics for one-dimensional continuous-Time Markov chains with polynomial transition rates

AU - Xu, Chuang

AU - Hansen, Mads Christian

AU - Wiuf, Carsten

N1 - Publisher Copyright: © The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - certain absorption

KW - Density-dependent continuous-Time Markov chains

KW - explosivity

KW - positive and null recurrence

KW - recurrence

KW - stationary and quasi-stationary distributions

KW - stochastic reaction networks

KW - transience

UR - http://www.scopus.com/inward/record.url?scp=85148608060&partnerID=8YFLogxK

U2 - 10.1017/apr.2022.20

DO - 10.1017/apr.2022.20

M3 - Journal article

AN - SCOPUS:85148608060

VL - 55

SP - 321

EP - 355

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 1

ER -