Some properties of stationary continuous state branching processes

We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton-Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general subcritical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

The study examines the genealogical tree of a stationary continuous state branching process with immigration, establishing distributions under various stable branching mechanisms and analyzing the number of individuals in the extant population who will produce descendants in the future. The transition rates of the associated Markov processes were computed to determine their characteristics.