{
  "id": "1503.04166",
  "version": "v1",
  "published": "2015-03-13T17:51:20.000Z",
  "updated": "2015-03-13T17:51:20.000Z",
  "title": "Equilibrium diffusion on the cone of discrete Radon measures",
  "authors": [
    "Diana Conache",
    "Yuri G. Kondratiev",
    "Eugene Lytvynov"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\mathbb K(\\mathbb R^d)$ denote the cone of discrete Radon measures on $\\mathbb R^d$. There is a natural differentiation on $\\mathbb K(\\mathbb R^d)$: for a differentiable function $F:\\mathbb K(\\mathbb R^d)\\to\\mathbb R$, one defines its gradient $\\nabla^{\\mathbb K} F $ as a vector field which assigns to each $\\eta\\in \\mathbb K(\\mathbb R^d)$ an element of a tangent space $T_\\eta(\\mathbb K(\\mathbb R^d))$ to $\\mathbb K(\\mathbb R^d)$ at point $\\eta$. Let $\\phi:\\mathbb R^d\\times\\mathbb R^d\\to\\mathbb R$ be a potential of pair interaction, and let $\\mu$ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on $\\mathbb R^d$. In particular, $\\mu$ is a probability measure on $\\mathbb K(\\mathbb R^d)$ such that the set of atoms of a discrete measure $\\eta\\in\\mathbb K(\\mathbb R^d)$ is $\\mu$-a.s.\\ dense in $\\mathbb R^d$. We consider the corresponding Dirichlet form $$ \\mathscr E^{\\mathbb K}(F,G)=\\int_{\\mathbb K(\\mathbb R^d)}\\langle\\nabla^{\\mathbb K} F(\\eta), \\nabla^{\\mathbb K} G(\\eta)\\rangle_{T_\\eta(\\mathbb K)}\\,d\\mu(\\eta). $$ Integrating by parts with respect to the measure $\\mu$, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If $d\\ge2$, there exists a conservative diffusion process on $\\mathbb K(\\mathbb R^d)$ which is properly associated with the Dirichlet form $\\mathscr E^{\\mathbb K}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-13T17:51:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "60G57"
    ],
    "keywords": [
      "discrete radon measures",
      "equilibrium diffusion",
      "corresponding gibbs perturbation",
      "conservative diffusion process",
      "corresponding dirichlet form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}