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Some Diffusion Processes Associated With Two Parameter Poisson-Dirichlet Distribution and Dirichlet Process

Abstract

The two parameter Poisson-Dirichlet distribution PD(α,θ)PD(\alpha,\theta) is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman's Poisson-Dirichlet distribution. The two parameter Dirichlet process Πα,θ,ν0\Pi_{\alpha,\theta,\nu_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(α,θ)PD(\alpha,\theta) and Πα,θ,ν0\Pi_{\alpha,\theta,\nu_0}. The methods used come from the theory of Dirichlet forms.

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