Column continuous transition functions
type: Item
instance of: scholarly article
title: Column continuous transition functions
zbMATH Open document ID: 1113.60078
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-12T00:00:00Z
review text: The author considers \(\mathbb{Z}^{+}\)-valued continuous-time Markov chains with column continuous transition functions. A transition function \(P(t) =(P_{ij}^{(t)})_{i,j\in \mathbb{Z}^{+}},\) \(t\geq 0\), is said to be column continuous if \(\lim_{t\rightarrow 0}P_{ij}(t) =\delta _{ij}\) uniformly in \(i\in \mathbb{Z}^{+}\) for any \(j\in \mathbb{Z}^{+}\). It is proved that such Markov chains have stable \(q\)-matrices. Next, there is a one-to-one correspondence between column continuous transition functions and increasing integrated semigroups on \(\ell _{\infty }\). Using the theory of integrated semigroups [see \textit{W. Arendt}, Isr. J. Math. 59, 327--352 (1987; Zbl 0637.44001)], the author gives necessary and sufficient conditions under which a (Feller) minimal transition function is column continuous in terms of its \(q\)-matrix \(Q\). He then constructs all column continuous transition functions with the same given conservative single-exit column bounded \(q\)-matrix \(Q\). Finally, several examples of Markov chains with column continuous transition functions are listed.
Mathematics Subject Classification ID: 60J27
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 5124093
zbMATH Keywords: continuous-time Markov chain
zbMATH Keywords: transition function
zbMATH Keywords: q-matrix, integrated semigroup
author: Yangrong Li
reviewed by: Marius Iosifescu
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.05.001
OpenAlex ID: W2087298750
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DOI: 10.1016/J.JMAA.2006.05.001
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description: scientific article; zbMATH DE number 5124093