Poisson-Dirichlet interval partition evolutions related to the Aldous diffusion

Seminar | April 4 | 3:10-4 p.m. | 1011 Evans Hall

 Matthias Winkel, University of Oxford

 Department of Statistics

We construct diffusions on a space of interval partitions of [0,1] that
are stationary with Poisson-Dirichlet laws. The processes of ranked
interval lengths of our partitions are diffusions introduced by Ethier and
Kurtz (1981) and Petrov (2009). Specifically, we decorate the jumps of a spectrally positive stable process with independent squared Bessel
excursions. In the spirit of Ray-Knight theorems, we form a process
indexed by level, in our case by extracting intervals from jumps crossing the level. We show that the fluctuating total interval lengths form another squared Bessel process of different dimension parameter. By interweaving two such constructions, we can match dimension parameters to equal -1. We normalize interval partitions and change time by total interval length in a de-Poissonisation procedure. These interval partition diffusions are a key ingredient to construct a diffusion on the space of real trees whose existence has been conjectured by David Aldous. This is joint work, partly in progress, with Noah Forman, Soumik Pal and Douglas Rizzolo.