{
  "id": "1801.09209",
  "version": "v1",
  "published": "2018-01-28T11:13:00.000Z",
  "updated": "2018-01-28T11:13:00.000Z",
  "title": "Nash inequality for Diffusion Processes Associated with Dirichlet Distributions",
  "authors": [
    "Feng-Yu Wang",
    "Weiwei Zhang"
  ],
  "comment": "20 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "For any $N\\ge 2$ and $\\alpha=(\\alpha_1,\\cdots, \\alpha_{N+1})\\in (0,\\infty)^{N+1}$, let $\\mu^{(N)}_{\\alpha}$ be the Dirichlet distribution with parameter $\\alpha$ on the set $\\Delta^{ (N)}:= \\{ x \\in [0,1]^N:\\ \\sum_{1\\le i\\le N}x_i \\le 1 \\}.$ The multivariate Dirichlet diffusion is associated with the Dirichlet form $${\\scr E}_\\alpha^{(N)}(f,f):= \\sum_{n=1}^N \\int_{ \\Delta^{(N)}} \\bigg(1-\\sum_{1\\le i\\le N}x_i\\bigg) x_n(\\partial_n f)^2(x)\\,\\mu^{(N)}_\\alpha(d x)$$ with Domain ${\\scr D}({\\scr E}_\\alpha^{(N)})$ being the closure of $C^1(\\Delta^{(N)})$. We prove the Nash inequality $$\\mu_\\alpha^{(N)}(f^2)\\le C {\\scr E}_\\alpha^{(N)}(f,f)^{\\frac p{p+1} }\\mu_\\alpha^{(N)} (|f|)^{\\frac 2 {p+1}},\\ \\ f\\in {\\scr D}({\\scr E}_\\alpha^{(N)}), \\mu_\\alpha^{(N)}(f)=0$$ for some constant $C>0$ and $p= (\\alpha_{N+1}-1)^+ +\\sum_{i=1}^N 1\\lor (2\\alpha_i),$ where the constant $p$ is sharp when $\\max_{1\\le i\\le N} \\alpha_i \\le 1/2$ and $\\alpha_{N+1}\\ge 1$. This Nash inequality also holds for the corresponding Fleming-Viot process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-28T11:13:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nash inequality",
      "dirichlet distribution",
      "diffusion processes",
      "multivariate dirichlet diffusion",
      "dirichlet form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}