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

Vidmar, Matija (author)

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 $q\in (0,\infty)$, 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.

• Language: English
• Keywords: Bienaymé–Galton–Watson process, branching, immigration, culling, harmonic function, first passage downwards, explosion, Laplace transform, factorization at the minimum, conditioning
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2022
• Number of pages: Str. 578-601
• Numbering: Vol. 94, no. 4
• PID: 20.500.12556/RUL-141243
• UDC: 519.217
• ISSN on article: 1744-2508
• DOI: 10.1080/17442508.2021.1963249
• COBISS.SI-ID: 73746947
• Publication date in RUL: 26.09.2022

Record is a part of a journal

• Title: Stochastics
• Publisher: Taylor & Francis
• ISSN: 1744-2508
• COBISS.SI-ID: 513295129

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: Bienaymé-Galton-Watsonov proces, razvejanje, imigracija, odbiranje, harmonična funkcija, prvi prehod navzdol, eksplozija, Laplaceova transformacija, faktorizacija na minimumu, pogojevanje

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: P1-0402
• Name: Matematična fizika