Column continuous transition functions
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.
scientific article; zbMATH DE number 5124093
scientific article