Brownian Motion and Models of Fragmentation and Coalescence

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Pitman and Yor are continuing their work on explicit descriptions of the distribution of various functionals of Brownian motion and related processes such as Brownian and Bessel bridges, meanders and excursions, and the development of novel methods for obtaining such descriptions. Stimulus for obtaining the exact distributions of ever more complicated Brownian functionals, and understanding various identities, has been provided by applications of Brownian excursion and Brownian bridge to the asymptotics of random combinatorial objects such as trees and mappings, connections with the theory of random partitions and random discrete distributions, and the applications of Brownian motion in mathematical finance. While a great number of explicit formulae are now known there remain many mysterious distributional coincidences of the kind which have in the past provided stimulus for the development of novel techniques and deeper understanding through such devices as path transformations and decompositions. This award will continue present lines of research into stochastic processes, particularly Brownian motion, random partitions, and coalescent processes. Brownian motion provides the foundation of the modern theory of continuous time random processes with continuous paths, and has applications in fields as diverse as physics and mathematical finance. Random partitions find applications to combinatorics, physics and genetics. Coalescent processes model random phenomena involving irreversible clustering or aggregation in a wide variety of contexts. This is fundamental research into the mathematical structure of stochastic processes. Progress in this direction enhances our understanding of these processes, and has potential for application in numerous fields of knowledge.