fragmentation, statistics, mathematics, Brownian motion, distribution theory, path transformations, stochastic processes, local time, excursions, random trees, random partitions, processes of coalescence
I have been interested in interfaces between the traditional theory of stochastic processes and other areas of mathematics, especially combinatorics. I have studied various random combinatorial objects, such as permutations, partitions, and trees, and how the asymptotic behaviour of such structures over a large number of elements can be described in probabilistic terms, most often involving Brownian motion and related processes. This has led to the study of various measure-valued and partition-valued Markov processes whose behaviour may be understood in terms of combinatorial constructions involving random trees. I am at present engaged in developing various ideas related to random partitions, random trees, irreversible processes of coalescence, and their time reversals which provide models for random splitting or fragmentation.
I view this line of research largely as pure mathematics, but mathematics of a concrete kind which is often motivated and influenced by applications. Stochastic models with a natural probabilistic structure typically turn up in different disguises in diverse fields. The study of their mathematical structure allows ideas and results developed in one context to be transferred to another.