Stochastic flows associated to coalescent processes. II: Stochastic differential equations

Abstract

We obtain precise information about the stochastic flows of bridges that are associated with the so-called Λ-coalescents. When the measure Λ gives no mass to 0, we prove that the flow of bridges is generated by a stochastic differential equation driven by a Poisson point process. On the other hand, the case Λ=δ0 of the Kingman coalescent gives rise to a flow of coalescing diffusions on the interval [0,1]. We also discuss a remarkable Brownian flow on the circle which has close connections with the Kingman coalescent.