Some harmonic functions for killed Markov branching processes with immigration and culling. Issue 4 (19th May 2022) - Vidmar, Matija
Abstract : For a continuous-time Bienaymé–Galton–Watson process, X, with immigration and culling, 0 as an absorbing state, call X q the process that results from killing X at rate q ∈ ( 0, ∞ ), followed by stopping it on extinction or explosion. Then an explicit identification of the relevant harmonic functions of X q allows to determine the Laplace transforms (at argument q ) of the first passage times downwards and of the explosion time for X . Strictly speaking, this is accomplished only when the killing rate q is sufficiently large (but always when the branching mechanism is not supercritical or if there is no culling). In particular, taking the limit q ↓ 0 (whenever possible) yields the passage downwards and explosion probabilities for X . A number of other consequences of these results are presented.