Anton Wakolbinger, Frankfurt: A decomposition of the Brownian excursion (Oberseminar Mathematische Stochastik)
Mittwoch, 29.01.2020 17:00 im Raum SRZ 117
We discuss a realizationwise correspondence between a Brownian excursion (conditioned to reach height one) and a triple consisting of
(1) the local time profile of the excursion,
(2) an array of independent time-homogeneous Poisson processes on the real line, and
(3) a fair coin tossing sequence,
where (2) and (3) encode the ordering by height respectively the left-right ordering of the subexcursions.
The three components turn out to be independent, with (1) giving a time change that is responsible for the time-homogeneity of the Poisson processes. By the Ray-Knight theorem, (1) is the excursion of a Feller branching diffusion; thus the metric structure associated with (2), which generates the so-called lookdown space, can be seen as representing the genealogy underlying the Feller branching diffusion. We will relate our approach also to earlier work of Aldous and Warren on Brownian excursions conditioned on their local time profile.
The lecture is based on joint work with Stephan Gufler and Goetz Kersting.