Infinitely Exchangeable Partition and Tree-valued Stochastic Processes

We discuss some aspects of infinitely exchangeable partition-valued stochastic processes. In particular, we introduce the the cut-and-paste process on the state space of partitions of the natural numbers whose sample paths differ from previously studied exchangeable coalescent (Kingman, 1982) and fragmentation (Bertoin, 2001) processes. Though it evolves differently, the cut-and-paste process possesses some of the same properties as its predecessors, including a unique equilibrium measure, associated measure-valued process, a Poisson point process construction and transition probabilities which can be described in terms of Kingman’s paintbox process. A parametric subfamily is related to the Ewens process (Ewens, 1972) and admits further extensions which may be natural in certain biological applications. We close by discussing some related, but disjoint, work on exchangeable tree-valued processes. Information about building access for persons with disabilities may be obtained in advance by calling Matt Johnston at 773.702-0541 or by email ([email protected]).