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

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 $q\in (0,\mathrm{\infty })$, 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.

Keywords:

• Bienaymé–Galton–Watson process
• branching
• immigration
• culling
• harmonic function
• first passage downwards
• explosion
• Laplace transform
• factorization at the minimum
• conditioning

2020 Mathematics Subject Classification:

• Primary: 60J80. Secondary: 60J50

Acknowledgments

The author thanks an anonymous Referee whose comments have helped to improve the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 We write expectations in the parlance of the theory of Markov processes: ${\mathsf{P}}_x[F;A]={\mathsf{E}}_{{\mathsf{P}}_x}\left[F{\mathbb{1}}_A\right]$, whenever it is defined.

2 By the qualifier ‘conservative’ we mean merely that the sum of each row of Q is zero.

3 If one takes for Ω a suitable canonical space, then for sure the conditions of the extension theorem are met. This one can do without affecting any distributional results.

Additional information

Funding

Financial support from the Slovenian Research Agency is acknowledged (programme No. P1-0402).