39 research outputs found
Some harmonic functions for killed Markov branching processes with immigration and culling
For a continuous-time Bienaym\'e-Galton-Watson process, , with immigration
and culling, as an absorbing state, call the process that results
from killing at rate , followed by stopping it on
extinction or explosion. Then an explicit identification of the relevant
harmonic functions of allows to determine the Laplace transforms (at
argument ) of the first passage times downwards and of the explosion time
for . Strictly speaking, this is accomplished only when the killing rate
is sufficiently large (but always when the branching mechanism is not
supercritical or if there is no culling). In particular, taking the limit
(whenever possible) yields the passage downwards and explosion
probabilities for . A number of other consequences of these results are
presented