Criteria for Feller transition functions
type: Item
instance of: scholarly article
title: Criteria for Feller transition functions
zbMATH Open document ID: 1184.60029
author: Jia Li
author: Yangrong Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-08-20T00:00:00Z
review text: The Feller property of a continuous-time Markov chain on the state space \(E=\mathbb{Z}_+=\{0,1,2,\dots\}\) with a transition function \(P(t)=(P_{ij}(t))\), \(t \geq 0\) and a \(q\)-matrix \(Q=(q_{ij})\) is considered; \(P(t)\) is assumed to be a \(Q\)-function, i.e. \(P_{ij}'(0)=q_{ij}\) (\(i,j \in E\)). The analysis is associated with individual states. Thus a state \(m\in E\) is said to be a Feller state if \(P_{im}(t) \to 0\) as \(i \to \infty\) for all \(t \geq 0\) and \(P(t)\) is proved to be the Feller transition function if and only if all states are Feller states. So, the Feller property describes the asymptotic behavior at the remote states. In the article some conditions for the minimal \(Q\)-function \(P(t)\) to be a Feller transition function, for a given \(q\)-matrix \(Q\), are presented: i) a sufficient condition that is stated explicitly in terms of the transition rates \(\{q_{ij}\}\); ii) some necessary and sufficient conditions in terms of properties of a system of equations (or inequalities) and in terms of the operator induced by the \(q\)-matrix. The criteria lead to some perturbation results which present a method to get some new Feller functions from the well-known ones. These results are applied to birth-death processes with killing, yielding some sufficient and some necessary conditions for the Feller property in terms of the transition rates.
reviewed by: Alex V. Kolnogorov
Mathematics Subject Classification ID: 60J27
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 5597542
zbMATH Keywords: continuous-time Markov chains
zbMATH Keywords: Feller transition functions
zbMATH Keywords: \(q\)-matrices
zbMATH Keywords: \(Q\)-functions
zbMATH Keywords: birth-death processes with killing
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.05.033
OpenAlex ID: W2056875384
cites work: Some remarks on the duality of continuous time markov chains
cites work: Applications of Feller-Reuter-Riley transition functions
cites work: Uniqueness criteria for continuous-time Markov chains with general transition structures
cites work: Existence, uniqueness, and constructions for stochastically monotone \(Q\)-processes
cites work: Quasi-stationary distributions for birth-death processes with killing
cites work: Birth-death processes with killing
cites work: Extinction Probability in A Birth-Death Process with Killing
cites work: On the existence of a quasistationary measure for a Markov chain
cites work: Contraction integrated semigroups and their application to continuous-time Markov chains
cites work: Dual and Feller-Reuter-Riley transition functions
cites work: Column continuous transition functions
cites work: Strongly monotone \(q\)-functions and a note on strong ergodicity of monotone \(q\)-functions
cites work: Explosive Markov branching processes: entrance laws and limiting behaviour
cites work: Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state
cites work: The Feller Property for Markov Semigroups on a Countable State Space
cites work: The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes
cites work: Q4141653
cites work: Stochastic comparability and dual \(Q\)-functions
cites work: Strong ergodicity of monotone transition functions
DOI: 10.1016/J.JMAA.2009.05.033
Recommended article: Applications of Feller-Reuter-Riley transition functions
Recommended article: Dual and Feller-Reuter-Riley transition functions
Recommended article: Q3537010
Recommended article: The transition function of a Fleming-Viot process
Recommended article: Q4034901
Recommended article: Functional properties of transition probability spaces
Recommended article: Q3811474
Recommended article: On the Existence of a Homogeneous Transition Function
Recommended article: Q5463015
description: scientific article; zbMATH DE number 5597542