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We construct a pair of related diffusions on a space of interval partitions of the unit interval $[0,1]$ that are stationary with the Poisson-Dirichlet laws with parameters (1/2,0) and (1/2,1/2) respectively. These are two particular cases of a general construction of such processes obtained by decorating the jumps of a spectrally positive Levy process with independent squared Bessel excursions. The processes of ranked interval lengths of our partitions are members of a two parameter family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009). The latter diffusions are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction is also a step towards describing a diffusion on the space of real trees whose existence has been conjectured by Aldous.
We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson-Dirichlet laws with parameters $(alpha,0)$ and $(alpha,alpha)$. The construction has two steps. The first is a general construction of
We give a pathwise construction of a two-parameter family of purely-atomic-measure-valued diffusions in which ranked masses of atoms are stationary with the Poisson-Dirichlet$(alpha,theta)$ distributions, for $alphain (0,1)$ and $thetage 0$. This res
We study interval partition diffusions with Poisson--Dirichlet$(alpha,theta)$ stationary distribution for parameters $alphain(0,1)$ and $thetage 0$. This extends previous work on the cases $(alpha,0)$ and $(alpha,alpha)$ and builds on our recent work
Consider a spectrally positive Stable($1+alpha$) process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning sizes varying during the lifetime. As for Crump-Mode-Jagers processes (with characteri
In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1. In the present paper, we represent this process in a space of interval partitions. We show that this is a member of a clas