Criteria for Feller transition functions
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.
scientific article; zbMATH DE number 5597542