On the boundary classification of Λ-Wright–Fisher processes with frequency-dependent selection
Annales Henri Lebesgue, Volume 6 (2023), pp. 493-539.


Keywords $\Lambda $-Wright–Fisher process, selection, $\Lambda $-coalescent, fragmentation-coalescence, duality, explosion, coming down from infinity, entrance boundary, regular boundary, continuous-time Markov chains


We construct extensions of the pure-jump Λ-Wright–Fisher processes with frequency-dependent selection (Λ-WF with selection) with different behaviors at their boundary 1. 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 1 and of the processes involved. They provide new information on these two classes of processes. Sufficient conditions are provided for boundary 1 to be an exit boundary or an entrance boundary. When the coalescence measure Λ and the selection mechanism verify some regular variation properties, conditions are found in order that the extended Λ-WF process with selection makes excursions out from the boundary 1 before getting absorbed at 0. In this case, 1 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.


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