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

The two parameter Poisson-Dirichlet distribution $PD(\alpha ,\theta )$ 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 ${\mathrm{\Pi }}_{\alpha ,\theta ,{\nu }_0}$ 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 $PD(\alpha ,\theta )$ and ${\mathrm{\Pi }}_{\alpha ,\theta ,{\nu }_0}$. The methods used come from the theory of Dirichlet forms.