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Register a new userA two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributions
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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.
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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 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.
The two parameter Poisson-Dirichlet distribution $PD(alpha,theta)$ is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingmans Poisson-Dirichlet distribution. The two parameter Dirichlet process $Pi_{alpha,theta, u_0}$ is the law of a pure atomic random measure with masses following the two parameter Poisson-Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures $PD(alpha,theta)$ and $Pi_{alpha,theta, u_0}$. The methods used come from the theory of Dirichlet forms.
The Gamma-Dirichlet structure corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the gamma process and vice versa. In this article, we begin with a brief review of existing results concerning the Gamma-Dirichlet structure. New results are obtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. The laws of the gamma process and the Dirichlet process are the respective reversible measures of the measure-valued branching diffusion with immigration and the Fleming-Viot process with parent independent mutation. We view the relation between these two classes of measure-valued processes as the dynamical Gamma-Dirichlet structure. Other results of this article include the derivation of the transition function of the Fleming-Viot process with parent independent mutation from the transition function of the measure-valued branching diffusion with immigration, and the establishment of the reversibility of the latter. One of these is related to an open problem by Ethier and Griffiths and the other leads to an alternative proof of the reversibility of the Fleming-Viot process.
Consider a reflecting diffusion in a domain in $R^d$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting process and the value of the drift vector has a product form. Moreover, the first component is the symmetrizing measure on the domain for the reflecting diffusion without inert drift, and the second component has a Gaussian distribution. We also consider processes where the drift is given in terms of the gradient of a potential.
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