A dynamic model for the two-parameter Dirichlet process

Let α = 1/2, θ > − 1/2, and ν0 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 = ℕ, we show that the bilinear form $$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{l} \mathcal{E}(F,G)=\frac{1}{2}{\int}_{\mathcal{P}_{1}(\mathbb{N})}\langle \nabla F(\mu),\nabla G(\mu)\rangle_{\mu} {\Pi}_{\alpha,\theta,\nu_{0}}(d\mu),\ \ F,G\in \mathcal{F},\\ \mathcal{F}=\{F(\mu)=f(\mu(1),\dots,\mu(d)):f\in C^{\infty}(\mathbb{R}^{d}), d\ge 1\} \end{array} \right. \end{array} $$ is closable on \(L^{2}(\mathcal {P}_{1}(\mathbb {N});{\Pi }_{\alpha ,\theta ,\nu _{0}})\) 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 \(\mathcal {P}_{1}(\mathbb {N})\) 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 $$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{l} \mathcal{E}(F,G)=\frac{1}{2}{\int}_{\mathcal{P}_{1}(S)}\langle \nabla F(\mu),\nabla G(\mu)\rangle_{\mu} {\Pi}_{\alpha,\theta,\nu_{0}}(d\mu),\ \ F,G\in \mathcal{F},\\ \mathcal{F}=\{F(\mu)=f(\langle \phi_{1},\mu\rangle,\dots,\langle \phi_{d},\mu\rangle): \phi_{i}\in B_{b}(S),1\le i\le d,f\in C^{\infty}(\mathbb{R}^{d}),d\ge 1\}. \end{array} \right. \end{array} $$ In particular, we prove the Mosco convergence of its projection forms.