Journal
STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES
Volume 94, Issue 4, Pages 578-601Publisher
TAYLOR & FRANCIS LTD
DOI: 10.1080/17442508.2021.1963249
Keywords
Bienayme-Galton-Watson process; branching; immigration; culling; harmonic function; first passage downwards; explosion; Laplace transform; factorization at the minimum; conditioning
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Funding
- Slovenian Research Agency [P1-0402]
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By identifying the relevant harmonic functions of X-q, it is possible to determine the Laplace transforms of the first passage times downwards and of the explosion time for X. This can only be done when the killing rate is sufficiently large, but is always achievable when the branching mechanism is not supercritical or if there is no 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.
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