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 $\Lambda $-Wright–Fisher process, selection, $\Lambda $-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.

References

[AN04] Athreya, Krishna B.; Ney, Peter E. Branching Processes, Dover Publications, 2004 (reprint of the 1972 original [Springer, New York; MR0373040]) | Zbl

[Ber96] Bertoin, Jean Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, 1996 | Zbl

[Ber04] Berestycki, Julien Exchangeable fragmentation-coalescence processes and their equilibrium measures, Electron. J. Probab., Volume 9 (2004), pp. 770-824 | MR | Zbl

[BLG03] Bertoin, Jean; Le Gall, Jean-François Stochastic flows associated with coalescent processes, Probab. Theory Relat. Fields, Volume 126 (2003) no. 2, pp. 261-288 | DOI | MR | Zbl

[BLG05] Bertoin, Jean; Le Gall, Jean-François Stochastic flows associated to coalescent processes. II. Stochastic differential equations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 41 (2005) no. 3, pp. 307-333 | DOI | Numdam | MR | Zbl

[BLW16] Baake, Ellen; Lenz, Ute; Wakolbinger, Anton The common ancestor type distribution of a $Λ$ -Wright Fisher process with selection and mutation, Electron. Commun. Probab., Volume 21 (2016), 59, 16 pages | MR | Zbl

[BOP20] Berzunza Ojeda, Gabriel; Pardo, Juan Carlos Branching processes with pairwise interactions (2020) (https://arxiv.org/abs/2009.11820v1)

[BP15] Bah, Boubacar; Pardoux, Etienne The $Λ$ -lookdown model with selection, Stochastic Processes Appl., Volume 125 (2015) no. 3, pp. 1089-1126 | MR | Zbl

[CR84] Cox, J. Theodore; Rösler, Uwe A duality relation for entrance and exit laws for Markov processes, Stochastic Processes Appl., Volume 16 (1984), pp. 141-156 | MR | Zbl

[CS18] Casanova, Adrián González; Spanò, Dario Duality and fixation in $Ξ$ -Wright-Fisher processes with frequency-dependent selection, Ann. Appl. Probab., Volume 28 (2018) no. 1, pp. 250-284 | MR | Zbl

[DK99] Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models, Ann. Probab., Volume 27 (1999) no. 1, pp. 166-205 | MR | Zbl

[DL12] Dawson, Donald A.; Li, Zenghu Stochastic equations, flows and measure-valued processes, Ann. Probab., Volume 40 (2012) no. 2, pp. 813-857 | MR | Zbl

[Don84] Doney, Ronald A. A note on some results of Schuh, J. Appl. Probab., Volume 21 (1984) no. 1, pp. 192-196 | DOI | MR | Zbl

[EG09] Etheridge, Alison M.; Griffiths, Robert C. A coalescent dual process in a Moran model with genic selection, Theor. Popul. Biol., Volume 75 (2009) no. 4, pp. 320-330 | DOI | Zbl

[EK86] Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence, Wiley Series in Probability and Mathematical Statistics: Probability and mathematical statistics, John Wiley & Sons, 1986 | DOI | Zbl

[Eth12] Etheridge, Alison M. Some Mathematical Models from Population Genetics: École d’Été de Probabilités de Saint-Flour XXXIX-2009, Lecture Notes in Mathematics, Springer, 2012 | Zbl

[Fou13] Foucart, Clément The impact of selection in the $Λ$ -Wright-Fisher model, Electron. Commun. Probab., Volume 18 (2013), 72, 10 pages | MR | Zbl

[Fou19] Foucart, Clément Continuous-state branching processes with competition: duality and reflection at infinity, Electron. J. Probab., Volume 24 (2019), 33, 38 pages | MR | Zbl

[Fou22] Foucart, Clément A phase transition in the coming down from infinity of simple exchangeable coalescence-fragmentation processes, Ann. Appl. Probab., Volume 32 (2022) no. 1, pp. 632-664 | MR | Zbl

[FZ22] Foucart, Clément; Zhou, Xiaowen On the explosion of the number of fragments in simple exchangeable fragmentation-coagulation processes, Ann. Inst. Henri Poincaré, Probab. Stat, Volume 58 (2022) no. 2, pp. 1182-1207 | MR | Zbl

[GCPP21] González Casanova, Adrián; Pardo, Juan Carlos; Pérez, José Luis Branching processes with interactions: the subcritical cooperative regime, Adv. Appl. Probab., Volume 53 (2021) no. 1, pp. 251-278 | DOI | MR | Zbl

[Gri14] Griffiths, Robert C. The $Λ$ -Fleming–Viot process and a connection with Wright-Fisher diffusion, Adv. Appl. Probab., Volume 46 (2014) no. 4, pp. 1009-1035 | DOI | MR | Zbl

[Har63] Harris, Theodore E. The Theory of Branching Processes, Grundlehren der Mathematischen Wissenschaften, 119, Springer, 1963 | DOI | MR | Zbl

[HP20] Hermann, Felix; Pfaffelhuber, Peter Markov branching processes with disasters: Extinction, survival and duality to $p$ -jump processes, Stochastic Processes Appl., Volume 130 (2020) no. 4, pp. 2488-2518 | DOI | MR | Zbl

[KN97] Krone, Stephen M.; Neuhauser, Claudia Ancestral processes with selection, Theor. Popul. Biol., Volume 51 (1997) no. 3, pp. 210-237 | DOI | Zbl

[Kol11] Kolokoltsov, Vassili N. Markov processes, semigroups and generators, de Gruyter Studies in Mathematics, 38, Walter de Gruyter, 2011 | Zbl

[KPRS17] Kyprianou, Andreas E.; Pagett, Steven W.; Rogers, Tim; Schweinsberg, Jason A phase transition in excursions from infinity of the fast fragmentation-coalescence process, Ann. Probab., Volume 45 (2017) no. 6A, pp. 3829-3849 | MR | Zbl

[KT81] Karlin, Samuel; Taylor, Howard M. A second course in stochastic processes, Academic Press Inc., 1981 | Zbl

[LP12] Li, Zenghu; Pu, Fei Strong solutions of jump-type stochastic equations, Electron. Commun. Probab., Volume 17 (2012) no. 13, 33 | MR | Zbl

[LT15] Limic, Vlada; Talarczyk, Anna Second-order asymptotics for the block counting process in a class of regularly varying Lambda-coalescents, Ann. Probab., Volume 43 (2015) no. 3, pp. 1419-1455 | Zbl

[Sch00] Schweinsberg, Jason A necessary and sufficient condition for the $Λ$ -coalescent to come down from infinity, Electron. Commun. Probab., Volume 5 (2000), pp. 1-11 | MR | Zbl

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

Metadata

Abstract

References

Cited by