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

## TL;DR

This paper explicitly characterizes harmonic functions for killed Markov branching processes with immigration and culling, enabling calculation of passage and explosion probabilities, especially for large killing rates.

## Contribution

It provides explicit harmonic functions for killed Markov branching processes with immigration and culling, facilitating the computation of passage and explosion probabilities.

## Key findings

- Explicit harmonic functions identified for large killing rates
- Laplace transforms of passage and explosion times derived
- Results applicable under non-supercritical or no culling conditions

## Abstract

For a continuous-time Bienaym\'e-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 $q\in (0,\infty)$, 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 $q\downarrow 0$ (whenever possible) yields the passage downwards and explosion probabilities for $X$. A number of other consequences of these results are presented.