The nested Kingman coalescent: Speed of coming down from infinity

Blancas, Airam; Rogers, Tim; Schweinsberg, Jason; Siri‐Jégousse, Arno

doi:10.1214/18-aap1440

Cited by 8 publications

(9 citation statements)

References 21 publications

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“…However, because of the domination by the standard coalescent, it is reasonable to conjecture that each species at time t still carries of the order of 1/t genes, so that the total number of genes at time t scales like 1/t 2 . This conjecture has been confirmed to hold in a recent work [9]. The starting point of this paper was the study of the distribution (rather than the total mass) of species masses at small times in the nested Kingman coalescent, for arbitrary initial conditions.…”

Section: Introduction and Informal Description Of The Main Resultssupporting

confidence: 55%

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Coagulation-transport equations and the nested coalescents

Lambert

Schertzer

2019

Probab. Theory Relat. Fields

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The nested Kingman coalescent describes the dynamics of particles (called genes) contained in larger components (called species), where pairs of species coalesce at constant rate and pairs of genes coalesce at constant rate provided they lie within the same species. We prove that starting from rn species, the empirical distribution of species masses (numbers of genes/n) at time t/n converges as n → ∞ to a solution of the deterministic coagulation-transport equationwhere ψ(x) = cx 2 , denotes convolution and a(t) = 1/(t + δ) with δ = 2/r. The most interesting case when δ = 0 corresponds to an infinite initial number of species. This equation describes the evolution of the distribution of species of mass x, where pairs of species can coalesce and each species' mass evolves likeẋ = −ψ(x). We provide two natural probabilistic solutions of the latter IPDE and address in detail the case when δ = 0. The first solution is expressed in terms of a branching particle system where particles carry masses behaving as independent continuous-state branching processes.The second one is the law of the solution to the following McKean-Vlasov equationwhere J is an inhomogeneous Poisson process with rate 1/(t + δ) and (v t ; t ≥ 0) is a sequence of independent rvs such that L(v t ) = L(x t ). We show that there is a unique solution to this equation and we construct this solution with the help of a marked Brownian coalescent point process. When ψ(x) = x γ , we show the existence of a self-similar solution for the PDE which relates when γ = 2 to the speed of coming down from infinity of the nested Kingman coalescent.

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Section: Introduction and Informal Description Of The Main Resultssupporting

confidence: 55%

“…Remark 1.9. The variable Υ also arises in the study [9] of the speed of coming down from infinity of the nested Kingman coalescent. However, Υ is not characterized in terms of an IDE or an excursion measure.…”

Section: Furthermentioning

confidence: 99%

Coagulation-transport equations and the nested coalescents

Lambert

Schertzer

2019

Probab. Theory Relat. Fields

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“…such that E ν s (dp, dµ)p 2 < ∞, 8) and such that for any array (g, s, c) ∈ C such that |g| = b, i s i = k and j c ij = l i ,…”

Section: Càdlàg Markov Process With Values In Pmentioning

confidence: 99%

“…In the case when the species tree is also distributed as a Kingman coalescent, the former two-type coalescent process is a Markov process as time runs backward, that we call the nested Kingman coalescent (or 'Kingman-in-Kingman') [8,11,28]. Our goal here is to display a much richer class of Markov models for trees within trees, called simple nested exchangeable coalescent (SNEC) processes, where multiple species lineages can merge into one single species lineage, and where simultaneously, within those merging species, multiple gene lineages can merge into one single gene lineage.…”

Section: Introductionmentioning

confidence: 99%

Trees within trees: simple nested coalescents

Blancas¹,

Duchamps²,

Lambert³

et al. 2018

Electron. J. Probab.

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We consider the compact space of pairs of nested partitions of N, where by analogy with models used in molecular evolution, we call "gene partition" the finer partition and "species partition" the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of Λ-coalescents to nested partitions. We characterize the law of SNEC processes as follows. In the absence of gene coalescences, species blocks undergo Λ-coalescent type events and in the absence of species coalescences, gene blocks lying in the same species block undergo i.i.d. Λ-coalescents. Simultaneous coalescence of the gene and species partitions are governed by an intensity measure ν s on (0, 1] × M 1 ([0, 1]) providing the frequency of species merging and the law in which are drawn (independently) the frequencies of genes merging in each coalescing species block. As an application, we also study the conditions under which a SNEC process comes down from infinity.

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“…The nested coalescent and its dual. The nested coalescent is an object introduced recently Blancas et al (2018), which has already received some attention Blancas et al (2019); Duchamps (2020); Lambert and Schertzer (2020). Its purpose is to integrate speciation events and individual reproduction in the same model, in order to be able to trace ancestry at the level of species.…”

Section: Moment Dualitymentioning

confidence: 99%