A coagulation-transport equation and the nested Kingman coalescent
speaker: Emmanuel Schertzer (Sorbonne Universite)abstract: Consider the following transport-coagulation equation \begin{equation} \forall t>0, \forall x\geq0, \ \ \partialt n (t,x) \ = \ \partialx (x\gamma n )(t,x) \ + \ \frac{1}{2} \int{0}x n(t,x-y) n(t,y) dy \ - \ n(t,x) \int{0}\infty n(t,y) dy. \end{equation} If we think of $n(t,x) dx$ as the ``number'' of clusters carrying a mass in an interval of size $dx$ around $x$ at time $t$, then the previous equation can be interpreted as the following dynamics: clusters coalesce at rate $1$, and the mass of each cluster is depleted at a rate proportional to $\mbox{mass}\gamma$. Our main motivation for studying the latter PDE is the nested coalescent model in which gene lineages are constrained by a phylogeny, i.e., ancestral lineages can only coalesce if they belong to the same species. In particular, when $\gamma=2$, we show that the latter PDE can be recovered the nested Kingman coalescent (where gene and species lineages are both described by a standard Kingman coalescent) at small time scales. In particular, we show that the existence of a self-similar solution for the PDE relates to the speed of coming down from infinity in the nested Kingman coalescent. I will also address some open problems related to the previous results.
This is joint work with A. Lambert
timetable:
Mon 1 Apr, 14:30 - 15:30, Aula Dini
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