Evolving coalescents

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Coalescent theory has become a topic of central importance at the interface of probability theory and population genetics. In recent years, coalescent structures evolving in time have gained particular interest, they serve as a model for the changes which genealogies of populations undergo in time. In the first funding period we have studied evolving coalescents as tree-valued processes, and analysed the asymptotic distribution of various functionals of evolving Beta coalescents. We found a method which allows to represent a rich class of functionals of Beta coalescents as stable Poisson integrals, simultaneously for the static and the evolving case. In the second funding period we will continue this line of research. We will elaborate on the method of Poisson integrals, with a particular focus on the internal length spectrum of Beta coalescents. Moreover we will zoom in on new functionals of the Bolthausen-Sznitman coalescent and extend our investigations of coalescent functionals to general Lambda-coalescents. Specifically we are interested in the external branches of extremal length and the size of the last merger. We also plan to investigate the symmetric excursion representation for genealogies under individual competition.