A dynamic model for the two-parameter Dirichlet process

Let $\alpha =1\mathrm{/}2$, $\theta >-1\mathrm{/}2$, 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 ${\mathrm{\Pi }}_{\alpha ,\theta ,{\nu }_0}$. 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 $L^2({\mathcal{P}}_1(\mathbb{N});{\mathrm{\Pi }}_{\alpha ,\theta ,{\nu }_0})$ and its closure $(\mathcal{E},D(\mathcal{E}))$ is a quasi-regular Dirichlet form. Hence $(\mathcal{E},D(\mathcal{E}))$ is associated with a diffusion process in ${\cal P}1($\mathbb{N}$)$whichistime-reversiblewiththestationarydistribution$\Pi{\alpha,\theta,\nu_0}$\mathrm{.}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.