On the boundary classification of $\Lambda $-Wright–Fisher processes with frequency-dependent selection

Foucart, Clément; Zhou, Xiaowen

doi:10.5802/ahl.170

Foucart, Clément; Zhou, Xiaowen

On the boundary classification of

Λ

-Wright–Fisher processes with frequency-dependent selection

Annales Henri Lebesgue, Volume 6 (2023), pp. 493-539.

Metadata

DOI10.5802/ahl.170

Keywords

Λ

-Wright–Fisher process, selection,

Λ

-coalescent, fragmentation-coalescence, duality, explosion, coming down from infinity, entrance boundary, regular boundary, continuous-time Markov chains

Abstract

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 $\infty$ 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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$On the boundary classification of <span class="mathjax-formula" data-tex="$\Lambda $"><math xmlns="http://www.w3.org/1998/Math/MathML" ><mi>Λ</mi></math></span>-Wright–Fisher processes with frequency-dependent selection - Annales Henri Lebesgue$

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