Stochastic flows associated to coalescent processes

Abstract

We study a class of stochastic flows connected to the coalescent processes that have been studied recently by Möhle, Pitman, Sagitov and Schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size. We define a bridge to be a right-continuous process (B(r),r ∈ [0,1]) with nondecreasing paths and exchangeable increments, such that B(0)=0 and B(1)=1. We show that flows of bridges are in one-to-one correspondence with the so-called exchangeable coalescents. This yields an infinite-dimensional version of the classical Kingman representation for exchangeable partitions of ℕ. We then propose a Poissonian construction of a general class of flows of bridges and identify the associated coalescents. We also discuss an important auxiliary measure-valued process, which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized Fleming-Viot process.