Brownian Motion and Random Partitions

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9703691 Pitman Recent work of the principal investigator and co-authors has made connections between random partitions, random discrete distributions, random trees, and the lengths of excursions of Brownian motion and other stochastic processes. The present project continues the study of these relations and their applications. Sheth's model for gravitational clustering of galaxies leads to problems related to the additive coalescent in which each pair of sets in a partition of a finite set is merging at a rate which is the sum of the sizes of the two sets. Asymptotics of this additive coalescent define a discrete measure valued process which can be constructed by cutting Aldous's continuum random tree at points of a suitable Poisson process. It is proposed to study this and other measure-valued Markov processes which model coalescent phenomena, and to look for novel applications of these processes. Brownian motion provides the doundation 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. Statistical models for physical processes of coalescence and fragmentation are of interest in a number of fields. The evolution of a partition of particles into clumps can be variously interpreted, for example, as a partition of atoms into molecules, a partition of molecules into polymers, or a partition of stars into galaxies. In a chemical setting, atoms coalesce to form molecules, and molecules coalesce to form polymers. In a cosmological setting, stars coalesce to form galaxies. The proposed research focuses on models for an irreversible process of coalescence of a partition of a continuum with finite total mass into a finite or countable collection of sub-masses, with conservation of mass.