Structural classification of continuous time Markov chains with applications

Xu, Chuang; Hansen, Mads Christian; Wiuf, Carsten

doi:10.1080/17442508.2021.2017937

Cited by 5 publications

(34 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. In light of the definition of essential SRNs and extinct SRNs, the results except the latter part of (i) follow directly from [33,Theorem 3.2]. Hence, it remains to show the latter part of (i).…”

Section: Structural Classification Of Srnsmentioning

confidence: 85%

“…0 , then its minimal set A is finite and A = B [33]. For a subset A ⊆ Z d , let #A be the cardinality of A, and span A be the smallest vector subspace over R containing A.…”

Section: Preliminariesmentioning

confidence: 99%

“…Not all subsets of Z d have a gcd (or even a common divisor), e.g., A = {(1, 2), (2, 1)}. Indeed, for A ⊆ Z d \ {0} non-empty, dim span (A) = 1 if and only if A has a common divisor, if and only if gcd(A) exists [33].…”

Section: Preliminariesmentioning

confidence: 99%

“…(ii) It is a direct result of [33,Theorem 3.8] According to [2], N ∪ T is an "extinction set" for all reactions. For further classification, we give the following definitions.…”

Section: Structural Classification Of Srnsmentioning

confidence: 99%

“…2 for a definition of the line segment [x, y] 1 , min 1 and max 1 ). Furthermore, define [33,Theorem 3.13,Corollary 3.15], and provides a detailed characterization of the relevant sets assuming the reaction network is not conservative. (In the statement below, x y , for x, y ∈ R d , refers to the proportionality constant of x and y, which exists due to the assumptions made, see Section 2.1.1.)…”

Section: Any Non-zero Vector In Smentioning

confidence: 99%

See 4 more Smart Citations

Classification and threshold dynamics of stochastic reaction networks

Wiuf¹,

Xu²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Stochastic reaction networks (SRNs) provide models of many realworld networks. Examples include networks in epidemiology, pharmacology, genetics, ecology, chemistry, and social sciences. Here, we model stochastic reaction networks by continuous time Markov chains (CTMCs) and derive new results on the decomposition of the ambient space N d 0 (with d ≥ 1 the number of species) into communicating classes. In particular, we propose to study (minimal) core networks of an SRN, and show that these characterize the decomposition of the ambient space.Special attention is given to one-dimensional mass-action SRNs (1-d stoichiometric subspace). In terms of (up to) four parameters, we provide sharp checkable criteria for various dynamical properties (including explosivity, recurrence, ergodicity, and the tail asymptotics of stationary or quasi-stationary distributions) of SRNs in the sense of their underlying CTMCs. As a result, we prove that all 1-d endotactic networks are non-explosive, and positive recurrent with an ergodic stationary distribution with Conley-Maxwell-Poisson (CMP)-like tail, provided they are essential. In particular, we prove the recently proposed positive recurrence conjecture in one dimension: Weakly reversible mass-action SRNs with 1-d stoichiometric subspaces are positive recurrent. The proofs of the main results rely on our recent work on CTMCs with polynomial transition rate functions.

show abstract

Section: Structural Classification Of Srnsmentioning

confidence: 85%

“…0 , then its minimal set A is finite and A = B [33]. For a subset A ⊆ Z d , let #A be the cardinality of A, and span A be the smallest vector subspace over R containing A.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…(ii) It is a direct result of [33,Theorem 3.8] According to [2], N ∪ T is an "extinction set" for all reactions. For further classification, we give the following definitions.…”

Section: Structural Classification Of Srnsmentioning

confidence: 99%

Section: Any Non-zero Vector In Smentioning

confidence: 99%

See 3 more Smart Citations

Classification and threshold dynamics of stochastic reaction networks

Wiuf¹,

Xu²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

Hansen

Wiuf

2022

Adv. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

The asymptotic tails of limit distributions of continuous-time Markov chains

Xu,

Hansen,

Wiuf

2023

Adv. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

This paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions (QSDs) of continuous-time Markov chains on subsets of the non-negative integers. Based on the so-called flux-balance equation, we establish identities for stationary measures and QSDs, which we use to derive tail asymptotics. In particular, for continuous-time Markov chains with asymptotic power law transition rates, tail asymptotics for stationary distributions and QSDs are classified into three types using three easily computable parameters: (i) super-exponential distributions, (ii) exponential-tailed distributions, and (iii) sub-exponential distributions. Our approach to establish tail asymptotics of stationary distributions is different from the classical semimartingale approach, and we do not impose ergodicity or moment bound conditions. In particular, the results also hold for explosive Markov chains, for which multiple stationary distributions may exist. Furthermore, our results on tail asymptotics of QSDs seem new. We apply our results to biochemical reaction networks, 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. Our approach, together with the identities, easily extends to discrete-time Markov chains.

show abstract

Structural classification of continuous time Markov chains with applications

Cited by 5 publications

References 18 publications

Classification and threshold dynamics of stochastic reaction networks

Classification and threshold dynamics of stochastic reaction networks

Full classification of dynamics for one-dimensional continuous-time Markov chains with polynomial transition rates

The asymptotic tails of limit distributions of continuous-time Markov chains

Contact Info

Product

Resources

About