# Prof. Dr. Martin Möhle, Universität Tübingen: "On a scaling limit for the Bolthausen-Sznitman coalescent"

Jul 07, 2017 from 04:15 PM to 05:15 PM

LMS 4 - Raum 424 - Kleiner Hörsaal

Abstract:

The first part of the talk is an introduction to the theory of coalescent processes. Coalescents are Markovian stochastic processes with state space the set of partitions of the natural numbers. During each transition blocks merge together according to certain rules. Coalescents with multiple mergers of blocks can be characterized be a measure Lambda on the unit interval. The most prominent example is the Kingman coalescent where Lambda is the Dirac measure at zero. In this case during each transition exactly two blocks merge together at rate one. Another important example is the Bolthausen-Sznitman coalescent, where Lambda is uniformly distributed on the unit interval. In this case mergers of more than two blocks occur with positive probability. In the second part of the talk the block counting process and the fixation line of the Bolthausen-Sznitman coalescent are analyzed. It is shown that these processes, properly scaled, converge in distribution with respect to the Skorohod topology to the Mittag-Leffler process and to Neveu's continuous-state branching process respectively as the initial state tends to infinity. Strong relations to Siegmund duality, Mehler semigroups and self-decomposability are pointed out. Extensions to a larger class of coalescents with dust that do not come down from infinity are indicated.