Poisson Dirichlet(α, θ)-Bridge Equations and Coagulation-Fragmentation Duality

Abstract: This paper derives distributional properties of a class of exchangeable bridges closely related to the Poisson-Dirichlet (α, θ) family of bridges. As demonstrated in previous works, stochastic equations based on Poisson-Dirichlet (α, θ) processes, play an important role in a variety of applications. Here we focus on their role in obtaining/identifying otherwise difficult distributional results for coagulation and fragmentation operators. In particular we show how these stochastic equations, as well as existing ones, lead to constructions of new large classes of coagulation and fragmentation operators that satisfy a duality property, and are otherwise easily manipulated. This class, builds on, and includes the duality relations developed in Pitman (33), Bertoin and Goldschmidt (3), and Dong, Goldschmidt and Martin (12)(DGM), which we can treat in a unified way. Among our results, [(i)]we identify a dual continuous time coagulation/fragmentation process which can be seen as a natural extension of the standard BolthausenSznitman/Ruelle processes. [(ii)] Identify a Markovian continuous timeinhomgeneous fragmentation process based on (PD(α, θ + t), t ≥ 0), with a tractable description of its transition distribution. In other words, this includes a Markovian continuous time Ewens fragmentation process. [(iii)] We also discuss some explicit quantities related to the operators discussed in DGM. Our exposition suggests an approach to obtain other dualities and related results via a calculus on bridges.