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.
Because of the independence of the three components, up to a time change the distribution of this genealogy does not change under a conditioning on the local time profile. This gives also a natural access to genealogies of continuum populations under competition, whose population size is modeled e.g. by the Fellerbranching diffusion with a logistic drift.
The lecture is based on joint work with Stephan Gufler and Goetz Kersting.