Full classification of dynamics for one-dimensional continuous-Time Markov chains with polynomial transition rates
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Full classification of dynamics for one-dimensional continuous-Time Markov chains with polynomial transition rates. / Xu, Chuang; Hansen, Mads Christian; Wiuf, Carsten.
I: Advances in Applied Probability, Bind 55, Nr. 1, 2023, s. 321-355.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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 -
ID: 338300720