Abstract: We construct a pair of related diffusions on a space of partitions of the unit interval whose stationary distributions are the complements of the zero sets of Brownian motion and Brownian bridge, respectively. These are two particular cases of a general construction of such processes obtained by decorating the jumps of a spectrally positive Levy process with independent squared Bessel excursions. The processes of ranked interval lengths of our partitions are members of a two parameter family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009) that are stationary with respect to the Poisson-Dirichlet distributions. Our construction is a step towards describing a diffusion on the space of real trees, stationary with respect to the law of the Brownian CRT, whose existence has been conjectured by Aldous.

Based on joint work with N. Forman, M. Winkel, and D. Rizzolo.

Bio: Professor Pal got his Ph.D. from Columbia University. After spending a two year post-doc at Cornell University he joined the University of Washington, Seattle. His research interest spans a variety of topics in probability: stochastic analysis, interacting particles, random matrices, random graphs and trees, optimal transport, and mathematical finance. He is a current Associate Editor of The Annals of Probability and a past Associate Editor of The Annals of Applied Probability.