Mathematisches Seminar

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

07.07.2017 von 16:15 bis 17:15

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.

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