Dynamics of continuous time Markov chains with applications to stochastic reaction networks

Chuang Xu, Mads Christian Hansen, Carsten Wiuf

Published 2019-09-27Version 1

This paper contributes to an in-depth study of properties of CTMCs on non-negative integer lattices, with particular interest in one-dimensional CTMCs with polynomial transitions rates. We study the classification of states for general CTMCs on the non-negative integer lattices, by characterizing the set of states (absorbing, trapping, escaping, positive irreducible components and quasi-irreducible components). For CTMCs on non-negative integers with polynomial transition rates, we provide threshold checkable criteria (in terms of parameters) for various dynamical properties-explosivity, recurrence versus transience, positive recurrence versus null recurrence, implosivity, as well as existence and non-existence of passage times; in particular, checkable sufficient conditions for exponential ergodicity of stationary and quasi-stationary distributions are obtained. Moreover, an identity for stationary measures is established and asymptotics of tails of stationary distributions are estimated. Similar identity as well as asymptotics are derived for quasi-stationary distributions. As a prominent application, we apply our results to stochastic (biochemical) reaction networks (SRNs). In particular, we show all weakly reversible mass-action reaction networks with one-dimensional stoichiometric subspaces not only are positive recurrent but have a unique exponentially ergodic stationary distribution with Conley-Maxwell-Poisson tail on every positive irreducible component, which confirms in one-dimensional case the recently proposed positive recurrence conjecture [D.F. Anderson and J. Kim, Some network conditions for positive recurrence of stochastically modeled reaction networks, SIAM J. Appl. Math., 78 (2018), 2692--2713.].