A dynamic model for the two-parameter Dirichlet process

Abstract

Let $\alpha=1/2$, $\theta>-1/2$, and $\nu_0$ be a probability measure on a type space $S$. In this paper, we investigate the stochastic dynamic model for the two-parameter Dirichlet process $\Pi_{\alpha,\theta,\nu_0}$. If $S=\mathbb{N}$, we show that the bilinear form \begin{eqnarray*} \left\{ \begin{array}{l} {\cal E}(F,G)=\frac{1}{2}\int_{{\cal P}_1(\mathbb{N})}\langle \nabla F(\mu),\nabla G(\mu)\rangle_{\mu} \Pi_{\alpha,\theta,\nu_0}(d\mu),\ \ F,G\in {\cal F},\\ {\cal F}=\{F(\mu)=f(\mu(1),\dots,\mu(d)):f\in C^{\infty}(\mathbb{R}^d), d\ge 1\} \end{array} \right. \end{eqnarray*} is closable on $L^2({\cal P}_1(\mathbb{N});\Pi_{\alpha,\theta,\nu_0})$ and its closure $({\cal E}, D({\cal E}))$ is a quasi-regular Dirichlet form. Hence $({\cal E}, D({\cal E}))$ is associated with a diffusion process in ${\cal P}_1(\mathbb{N})$ which is time-reversible with the stationary distribution $\Pi_{\alpha,\theta,\nu_0}$. If $S$ is a general locally compact, separable metric space, we discuss properties of the model \begin{eqnarray*} \left\{ \begin{array}{l} {\cal E}(F,G)=\frac{1}{2}\int_{{\cal P}_1(S)}\langle \nabla F(\mu),\nabla G(\mu)\rangle_{\mu} \Pi_{\alpha,\theta,\nu_0}(d\mu),\ \ F,G\in {\cal F},\\ {\cal F}=\{F(\mu)=f(\langle \phi_1,\mu\rangle,\dots,\langle \phi_d,\mu\rangle): \phi_i\in B_b(S),1\le i\le d,f\in C^{\infty}(\mathbb{R}^d),d\ge 1\}. \end{array} \right. \end{eqnarray*} In particular, we prove the Mosco convergence of its projection forms.