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# Coagulation-transport equations and the nested coalescents

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## 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 $$n\rightarrow \infty$$ to a solution of the deterministic coagulation-transport equation
\begin{aligned} \partial _t d \ = \ \partial _x ( \psi d ) \ + \ a(t)\left( d\,\star \,d - d \right) , \end{aligned}
where $$\psi (x) = cx^2$$, $$\star$$ denotes convolution and $$a(t)= 1/(t+\delta )$$ with $$\delta =2/r$$. The most interesting case when $$\delta =0$$ 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 like $$\dot{x} = -\psi (x)$$. We provide two natural probabilistic solutions of the latter IPDE and address in detail the case when $$\delta =0$$. 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 equation
\begin{aligned} dx_t \ = \ - \psi (x_t) \,dt \ + \ v_t\,\Delta J_t \end{aligned}
where J is an inhomogeneous Poisson process with rate $$1/(t+\delta )$$ and $$(v_t; t\ge 0)$$ is a sequence of independent random variables such that $${{\mathcal {L}}}(v_t) = {{\mathcal {L}}}(x_t)$$. 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 $$\psi (x)=x^\gamma$$, we show the existence of a self-similar solution for the PDE which relates when $$\gamma =2$$ to the speed of coming down from infinity of the nested Kingman coalescent.

## Keywords

Kingman coalescent Smoluchowski equation McKean–Vlasov equation Degenerate PDE PDE probabilistic solution Hydrodynamic limit Entrance boundary Empirical measure Coalescent point process Continuous-state branching process Phylogenetics

## Mathematics Subject Classification

Primary 60K35 Secondary 35Q91 35R09 60G09 60B10 60G55 60G57 60J25 60J75 60J80 62G30 92D15

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## Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

## Authors and Affiliations

• Amaury Lambert
• 1
• 2
• Emmanuel Schertzer
• 1
• 2
1. 1.Laboratoire de Probabilités, Statistique et ModélisationSorbonne Université, CNRS UMR 8001ParisFrance
2. 2.Centre Interdisciplinaire de Recherche en BiologieCollège de France, CNRS UMR 7241ParisFrance