Uniqueness and decay properties of Markov branching processes with disasters

Abstract

In this paper we discuss the decay properties of Markov branching processes with disasters, including the decay parameter, invariant measures, and quasistationary distributions. After showing that the corresponding q-matrix Q is always regular and, thus, that the Feller minimal Q-process is honest, we obtain the exact value of the decay parameter λ_C. We show that the decay parameter can be easily expressed explicitly. We further show that the Markov branching process with disaster is always λ_C-positive. The invariant vectors, the invariant measures, and the quasidistributions are given explicitly.