Some harmonic functions for killed Markov branching processes with immigration and culling. Issue 4 (19th May 2022)

• Record Type: Journal Article
• Authors: Vidmar, Matija
• Is Part Of: Stochastics. Volume 94:Issue 4(2022)
• Page Start: 578
• Page End: 601
• Publication Date: 2022-05-19
• Subjects: Bienaymé–Galton–Watson process -- branching -- immigration -- culling -- harmonic function -- first passage downwards -- explosion -- Laplace transform -- factorization at the minimum -- conditioning; Primary: 60J80. Secondary: 60J50; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/17442508.2021.1963249
• Languages: English
• ISSNs: 1744-2508

Abstract

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.