{
  "id": "1706.06146",
  "version": "v1",
  "published": "2017-06-19T19:18:46.000Z",
  "updated": "2017-06-19T19:18:46.000Z",
  "title": "A dynamic model for the two-parameter Dirichlet process",
  "authors": [
    "Shui Feng",
    "Wei Sun"
  ],
  "comment": "19 pages",
  "categories": [
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-19T19:18:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G57",
      "60H30"
    ],
    "keywords": [
      "two-parameter dirichlet process",
      "stochastic dynamic model",
      "quasi-regular dirichlet form",
      "separable metric space",
      "type space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}