Some harmonic functions for killed Markov branching processes with immigration and culling

For a continuous-time Bienayme-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 is an element of (0, infinity) 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 down arrow 0 (whenever possible) yields the passage downwards and explosion probabilities forX. A number of other consequences of these results are presented.