Some diffusion processes associated with two parameter Poisson–Dirichlet distribution and Dirichlet process

The two parameter Poisson–Dirichlet distribution PD(α, θ) 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_{\alpha,\theta,\nu_0}}$$\end{document} 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) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_{\alpha,\theta,\nu_0}}$$\end{document}. The methods used come from the theory of Dirichlet forms.