[Sur la classification des points frontières des processus de -Wright–Fisher avec sélection dépendante de la fréquence]
We construct extensions of the pure-jump -Wright–Fisher processes with frequency-dependent selection (-WF with selection) with different behaviors at their boundary . Those processes satisfy some duality relationships with the block counting process of simple exchangeable fragmentation-coagulation processes (EFC processes). One-to-one correspondences are established between the nature of the boundaries and of the processes involved. They provide new information on these two classes of processes. Sufficient conditions are provided for boundary to be an exit boundary or an entrance boundary. When the coalescence measure and the selection mechanism verify some regular variation properties, conditions are found in order that the extended -WF process with selection makes excursions out from the boundary before getting absorbed at . In this case, is a transient regular reflecting boundary. This corresponds to a new phenomenon for the deleterious allele, which can be carried by the whole population for a set of times of zero Lebesgue measure, before vanishing in finite time almost surely.
Nous construisons des extensions des processus de -Wright–Fisher de saut pur avec sélection dépendante de la fréquence (-WF avec sélection) présentant différents comportement en leur point frontière . Ces processus satisfont des relations de dualité avec le processus du nombre de blocs des processus de fragmentation-coagulation échangeables simples. Des correspondances biunivoques entre les natures des frontières et des processus en question sont établies. Elles fournissent de nouvelles informations sur ces deux classes de processus. Des conditions suffisantes sont données pour que la frontière soit un point de sortie ou un point d’entrée. Lorsque la mesure et la fonction de sélection vérifient des propriétés de variations régulières, des conditions sont trouvées de sorte que le processus de -WF avec sélection étendu fasse des excursions en dehors de la frontière avant d’être absorbée en . Dans ce cas, est un point régulier réfléchissant et transient. Cela correspond à un nouveau phénomène pour l’allèle délétère, qui peut être porté par toute la population pendant un ensemble de temps de mesure de Lebesgue nulle, avant de disparaître en temps fini presque sûrement.
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Keywords: $\Lambda $-Wright–Fisher process, selection, $\Lambda $-coalescent, fragmentation-coalescence, duality, explosion, coming down from infinity, entrance boundary, regular boundary, continuous-time Markov chains
Foucart, Clément 1 ; Zhou, Xiaowen 2
CC-BY 4.0
@article{AHL_2023__6__493_0,
author = {Foucart, Cl\'ement and Zhou, Xiaowen},
title = {On the boundary classification of $\Lambda ${-Wright{\textendash}Fisher} processes with frequency-dependent selection},
journal = {Annales Henri Lebesgue},
pages = {493--539},
year = {2023},
publisher = {\'ENS Rennes},
volume = {6},
doi = {10.5802/ahl.170},
language = {en},
url = {https://www.numdam.org/articles/10.5802/ahl.170/}
}
TY - JOUR AU - Foucart, Clément AU - Zhou, Xiaowen TI - On the boundary classification of $\Lambda $-Wright–Fisher processes with frequency-dependent selection JO - Annales Henri Lebesgue PY - 2023 SP - 493 EP - 539 VL - 6 PB - ÉNS Rennes UR - https://www.numdam.org/articles/10.5802/ahl.170/ DO - 10.5802/ahl.170 LA - en ID - AHL_2023__6__493_0 ER -
%0 Journal Article %A Foucart, Clément %A Zhou, Xiaowen %T On the boundary classification of $\Lambda $-Wright–Fisher processes with frequency-dependent selection %J Annales Henri Lebesgue %D 2023 %P 493-539 %V 6 %I ÉNS Rennes %U https://www.numdam.org/articles/10.5802/ahl.170/ %R 10.5802/ahl.170 %G en %F AHL_2023__6__493_0
Foucart, Clément; Zhou, Xiaowen. On the boundary classification of $\Lambda $-Wright–Fisher processes with frequency-dependent selection. Annales Henri Lebesgue, Tome 6 (2023), pp. 493-539. doi: 10.5802/ahl.170
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