Criteria for Feller transition functions

Li, Yangrong; Li, Jia

doi:10.1016/j.jmaa.2009.05.033

Cited by 7 publications

(6 citation statements)

References 19 publications

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“…We note that Li and Li [97,Theorem 6.2(i)] show that if, under the conditions of this theorem, S abs is finite is replaced by the weaker condition lim i→∞ a i = 0, then divergence of (41) is sufficient for asymptotic remoteness, and hence, by Theorem 9, for the existence of a quasi-stationary distribution.…”

Section: Birth-death Processes With Killingmentioning

confidence: 92%

Quasi-Stationary Distributions

2010

Constructive Computation in Stochastic Models With Applications

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This paper contains a survey of results related to quasi-stationary distributions, which arise in the setting of stochastic dynamical systems that eventually evanesce, and which may be useful in describing the long-term behaviour of such systems before evanescence. We are concerned mainly with continuous-time Markov chains over a finite or countably infinite state space, since these processes most often arise in applications, but will make reference to results for other processes where appropriate. Next to giving an historical account of the subject, we review the most important results on the existence and identification of quasi-stationary distributions for general Markov chains, and give special attention to birth-death processes and related models. Results on the question of whether a quasi-stationary distribution, given its existence, is indeed a good descriptor of the long-term behaviour of a system before evanescence, are reviewed as well. The paper is concluded with a summary of recent developments in numerical and approximation methods.

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Section: Birth-death Processes With Killingmentioning

confidence: 92%

Quasi-Stationary Distributions

2010

Constructive Computation in Stochastic Models With Applications

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“…Explosion might occur for ∂ = ∅ with probability less than one for CTMCs with nonpolynomial transition rates [25]. Reuter's criterion and its generalizations provide necessary and sufficient conditions for explosivity (with positive probability) for general CTMCs in terms of convergence or divergence of a series [10,22,30], but they are not easy to check. In comparison, for CTMCs with polynomial transition rates, Theorem 3.1 provides an explicit and checkable necessary and sufficient condition.…”

Section: 1mentioning

confidence: 99%

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

Xu¹,

Hansen²,

Wiuf³

2020

Preprint

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This paper studies the dynamics of continuous time Markov chains with possibly unbounded jump sizes on the non-negative integers with polynomial transition rate functions. Such stochastic processes are abundant in applications, in particular in biology. We provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, certain absorption, positive/null recurrence, implosivity, and existence and non-existence of moments of hitting times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained, and the few gap cases are identified and well-illustrated by examples. Subtle differences in conditions for different dynamical properties are revealed in terms of examples. 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.

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“…In contrast, this might occur with non-polynomial transition rates [36]. Reuter's criterion and its generalizations provide necessary and sufficient conditions for explosivity for general CTMCs [33,45], though they are not easy to check. In comparison, for CTMCs with polynomial transition rates (and d = 1), Theorem 4.1 provides an explicit necessary and sufficient condition.…”

Section: Then Y T Explode If and Only If One Of The Following Conditi...mentioning

confidence: 99%

Dynamics of continuous time Markov chains with applications

Xu,

Hansen,

Wiuf

2019

Preprint

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This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes are abundant in applications, in particular within biology. We study the classification of states for general CTMCs on the non-negative integer lattices, by characterizing the set of absorbing states (similarly, trapping, escaping, positive irreducible components and quasi-irreducible components). For CTMCs on non-negative integers with polynomial transition rates, we provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, positive/null recurrence, implosivity, and existence and non-existence of passage times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained. Moreover, an identity for stationary measures is established and tail asymptotics for stationary distributions is given. A similar identity as well as asymptotics is derived for quasi-stationary distributions. Finally, we apply our results to stochastic reaction networks.

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Criteria for Feller transition functions

Cited by 7 publications

References 19 publications

Quasi-Stationary Distributions

Quasi-Stationary Distributions

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

Dynamics of continuous time Markov chains with applications

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