Uniqueness, extinction and explosivity of generalised Markov branching processes with pairwise interaction
The paper deals with Markov branching processes with pairwise interaction. These processes are certain continuous-time Markov chains with state space \(\mathbb{Z}_{\geq 0}\). They are defined via their \(q\)-matrix \(Q\) which in turn is defined in terms of two sequences, \((b_n)_{n \geq 0}\) and \((w_n)_{n \geq 0}\). The former basically governs the branching law whereas the latter governs the degree of interaction. It is shown that if the \(q\)-matrix \(Q\) is essentially-explosive (a notion that is defined in the paper), then the process is honest iff the mean death rate is greater than or equal to the mean birth rate. In the sub-explosive regime, the process is shown to be honest except in special cases. Further, a uniqueness criterion is provided. Finally, extinction times and explosivity are investigated.
scientific article; zbMATH DE number 5800284
scientific article