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 -