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Register a new userDiffusions on a space of interval partitions: Poisson-Dirichlet stationary distributions
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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 interval partition processes obtained previously, by decorating the jumps of a Levy process with independent excursions. Here, we focus on the second step, which requires explicit transition kernels and what we call pseudo-stationarity. This allows us to study processes obtained from the original construction via scaling and time-change. In a sequel paper, we establish connections to diffusions on decreasing sequences 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 resolving longstanding conjectures by Feng and Sun on measure-valued Poisson-Dirichlet diffusions, and by Aldous on a continuum-tree-valued diffusion.
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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 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 resolves a conjecture of Feng and Sun (2010). We build on our previous work on $(alpha,0)$- and $(alpha,alpha)$-interval partition evolutions. Indeed, we first extract a self-similar superprocess from the levels of stable processes whose jumps are decorated with squared Bessel excursions and distinct allelic types. We complete our construction by time-change and normalisation to unit mass. In a companion paper, we show that the ranked masses of the measure-valued processes evolve according to a two-parameter family of diffusions introduced by Petrov (2009), extending work of Ethier and Kurtz (1981). These ranked-mass diffusions arise as continuum limits of up-down Markov chains on Chinese restaurant processes.
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 on measure-valued diffusions. We work on spaces of interval partitions with $alpha$-diversity. These processes can be viewed as diffusions on the boundary of a branching graph of integer compositions. The additional order and diversity structure of such interval partitions is essential for applications to continuum random tree models such as stable CRTs and limit structures of other regenerative tree growth processes, where intervals correspond to masses of spinal subtrees (or spinal bushes) in spinal order and diversities give distances between any two spinal branch points. We further show that our processes can be extended to enter continuously from the Hausdorff completion of our state space and that, in contrast to the measure-valued setting, these extensions are Feller processes.
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 characteristics), we consider for each level the collection of individuals alive. We arrange their sizes at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable($1+alpha$) process, this yields new theorems of Ray-Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson--Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.
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 class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel using a completely different construction from spectrally positive stable Levy processes with index between 1 and 2 and with jumps marked by squared Bessel excursions of a corresponding dimension between $-2$ and 0.
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