Structural classification of continuous time Markov chains with applications
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Structural classification of continuous time Markov chains with applications. / Xu, Chuang; Hansen, Mads Christian; Wiuf, Carsten.
I: Stochastics, Bind 94, Nr. 7, 2022, s. 1003-1030.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Structural classification of continuous time Markov chains with applications
AU - Xu, Chuang
AU - Hansen, Mads Christian
AU - Wiuf, Carsten
N1 - Funding Information: The work was initiated with most part of it done when the first author was at the University of Copenhagen. The authors thank the editors' and referees' comments which helped improve the presentation of the paper. The authors acknowledge the support from The Erwin Schrödinger Institute (ESI) for the workshop on “Advances in Chemical Reaction Network Theory”. Publisher Copyright: © 2021 Informa UK Limited, trading as Taylor & Francis Group.
PY - 2022
Y1 - 2022
N2 - This paper is motivated by examples from stochastic reaction network theory. The Q-matrix of a stochastic reaction network can be derived from the reaction graph, an edge-labelled directed graph encoding the jump vectors of an associated continuous time Markov chain on the invariant space (Formula presented.). An open question is how to decompose the space (Formula presented.) into neutral, trapping, and escaping states, and open and closed communicating classes, and whether this can be done from the reaction graph alone. Such general continuous time Markov chains can be understood as natural generalizations of birth-death processes, incorporating multiple different birth and death mechanisms. We characterize the structure of (Formula presented.) imposed by a general Q-matrix generating continuous time Markov chains with values in (Formula presented.), in terms of the set of jump vectors and their corresponding transition rate functions. Thus the setting is not limited to stochastic reaction networks. Furthermore, we define structural equivalence of two Q-matrices, and provide sufficient conditions for structural equivalence. Examples are abundant in applications. We apply the results to stochastic reaction networks, a Lotka-Volterra model in ecology, the EnvZ-OmpR system in systems biology, and a class of extended branching processes, none of which are birth-death processes.
AB - This paper is motivated by examples from stochastic reaction network theory. The Q-matrix of a stochastic reaction network can be derived from the reaction graph, an edge-labelled directed graph encoding the jump vectors of an associated continuous time Markov chain on the invariant space (Formula presented.). An open question is how to decompose the space (Formula presented.) into neutral, trapping, and escaping states, and open and closed communicating classes, and whether this can be done from the reaction graph alone. Such general continuous time Markov chains can be understood as natural generalizations of birth-death processes, incorporating multiple different birth and death mechanisms. We characterize the structure of (Formula presented.) imposed by a general Q-matrix generating continuous time Markov chains with values in (Formula presented.), in terms of the set of jump vectors and their corresponding transition rate functions. Thus the setting is not limited to stochastic reaction networks. Furthermore, we define structural equivalence of two Q-matrices, and provide sufficient conditions for structural equivalence. Examples are abundant in applications. We apply the results to stochastic reaction networks, a Lotka-Volterra model in ecology, the EnvZ-OmpR system in systems biology, and a class of extended branching processes, none of which are birth-death processes.
KW - birth-death processes
KW - extinction
KW - persistence
KW - positive irreducible components
KW - Q-matrix
KW - quasi irreducible components
KW - stochastic reaction networks
KW - structural equivalence
UR - http://www.scopus.com/inward/record.url?scp=85121778359&partnerID=8YFLogxK
U2 - 10.1080/17442508.2021.2017937
DO - 10.1080/17442508.2021.2017937
M3 - Journal article
AN - SCOPUS:85121778359
VL - 94
SP - 1003
EP - 1030
JO - Stochastics: An International Journal of Probability and Stochastic Processes
JF - Stochastics: An International Journal of Probability and Stochastic Processes
SN - 1744-2508
IS - 7
ER -
ID: 289099431