Uniqueness and extinction properties of generalised Markov branching processes

This paper focuses on the basic problems regarding uniqueness and extinction properties for generalised Markov branching processes. The uniqueness criterion is firstly established and a differential–integral equation satisfied by the transition functions of such processes is derived. The extinction probability is then obtained. A closed form is presented for both the mean extinction time and the conditional mean extinction time. It turns out that these important quantities are closely related to the elementary gamma function.

Probabilistic approach in weighted Markov branching processes

This note provides a new probabilistic approach in discussing the weighted Markov branching process (WMBP) which is a natural generalisation of the ordinary Markov branching process. Using this approach, some important characteristics regarding the hitting times of such processes can be easily obtained. In particular, the closed forms for the mean extinction time and conditional mean extinction time are presented. The explosion behaviour of the process is investigated and the mean explosion time is derived. The mean global holding time and the mean total survival time are also obtained. The close link between these newly developed processes and the well-known compound Poisson processes is investigated. It is revealed that any weighted Markov branching process (WMBP) is a random time change of a compound Poisson process.