Coagulation-transport equations and the nested coalescents

Abstract

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.

Trees within trees: simple nested coalescents

“…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.…”

“…From a mathematical point of view as well, SNEC processes open up the door to many possible new investigations. For example some of us are currently studying the speed of coming down from infinity of SNEC processes [8,28] as well as similar extensions [15] to fragmentation processes [4]. It will be interesting to investigate how the nested trees generated by SNEC processes can be cast in the frameworks of multilevel measure-valued processes [7,11] and flows of bridges [5,6] as well as of exchangeable combs [20,26].…”

Particle systems with coordination

“…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.…”

An Exactly Solvable Continuous-Time Derrida–Retaux Model

“…However, we were not able to find references in the PDE literature to partial differential equations exhibiting a behavior similar to (1.10). Nevertheless, it has recently gained focus in the probability literature, for example in the recent article of Lambert and Schertzer [19], where a PDE of the form…”