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 $n\to\infty$ to a solution of the deterministic coagulation-transport equation $$ \partial_t d \ = \ \partial_x ( \psi d ) \ + \ a(t)\left(d\star d - d \right), $$ 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 $$ dx_t \ = \ - \psi(x_t) \,dt \ + \ v_t\,\Delta J_t $$ where $J$ is an inhomogeneous Poisson process with rate $1/(t+\delta)$ and $(v_t; t\geq0)$ is a sequence of independent rvs 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.