6 research outputs found
Stationary distributions and condensation in autocatalytic reaction networks
We investigate a broad family of stochastically modeled reaction networks by looking at their stationary distributions. Most known results on stationary distributions assume weak reversibility and zero deficiency. We first explicitly give product-form stationary distributions for a class of mostly non-weakly-reversible autocatalytic reaction networks of arbitrary deficiency. We provide examples of interest in statistical mechanics (inclusion process), life sciences, and robotics (collective decision making in ant and robot swarms). The product-form nature of the stationary distribution then enables the study of condensation in particle systems that are generalizations of the inclusion process
An algebraic approach to product-form stationary distributions for some reaction networks
Exact results for product-form stationary distributions of Markov chains are
of interest in different fields. In stochastic reaction networks (CRNs),
stationary distributions are mostly known in special cases where they are of
product-form. However, there is no full characterization of the classes of
networks whose stationary distributions have product-form. We develop an
algebraic approach to product-form stationary distributions in the framework of
CRNs. Under certain hypotheses on linearity and decomposition of the state
space for conservative ergodic CRNs, this gives sufficient and necessary
algebraic conditions for product-form stationary distributions. Correspondingly
we obtain a semialgebraic subset of the parameter space that captures rates
where, under the corresponding hypotheses, CRNs have product-form. We employ
the developed theory to CRNs and some models of statistical mechanics, besides
sketching the pertinence in other models from applied probability.Comment: Accepted for publication in SIAM Journal on Applied Dynamical System
An algebraic approach to product-form stationary distributions for some reaction networks
Exact results for product-form stationary distributions of Markov chains are of interest in different fields. In stochastic reaction networks (CRNs), stationary distributions are mostly known in special cases where they are of product-form. However, there is no full characterization of the classes of networks whose stationary distributions have product-form. We develop an algebraic approach to product-form stationary distributions in the framework of CRNs. Under certain hypotheses on linearity and decomposition of the state space for conservative CRNs, this gives sufficient and necessary algebraic conditions for product-form stationary distributions. Correspondingly, we obtain a semialgebraic subset of the parameter space that captures rates where, under the corresponding hypotheses, CRNs have product-form. We employ the developed theory to CRNs and some models of statistical mechanics, besides sketching the pertinence in other models from applied probability.The work of the first author was supported by the European Union's Horizon 2020 Research and
Innovation Programme under the Marie Sklodowska-Curie IF grant 794627. The work of the second author was
supported by Swiss National Science Foundations Early Postdoctoral Mobility grant P2FRP2 188023.Publicad
The asymptotic tails of limit distributions of continuous time Markov chains
This paper investigates tail asymptotics of stationary distributions and
quasi-stationary distributions of continuous-time Markov chains on a subset of
the non-negative integers. A new identity for stationary measures is
established. In particular, for continuous-time Markov chains with asymptotic
power-law transition rates, tail asymptotics for stationary distributions are
classified into three types by three easily computable parameters: (i)
Conley-Maxwell-Poisson distributions (light-tailed), (ii) exponential-tailed
distributions, and (iii) heavy-tailed distributions. Similar results are
derived for quasi-stationary distributions. The approach to establish tail
asymptotics is different from the classical semimartingale approach. We apply
our results to biochemical reaction networks (modeled as continuous-time Markov
chains), a general single-cell stochastic gene expression model, an extended
class of branching processes, and stochastic population processes with bursty
reproduction, none of which are birth-death processes