{
  "id": "math/0306252",
  "version": "v1",
  "published": "2003-06-17T10:52:19.000Z",
  "updated": "2003-06-17T10:52:19.000Z",
  "title": "Glauber dynamics of continuous particle systems",
  "authors": [
    "Yu. Kondratiev",
    "E. Lytvynov"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "This paper is devoted to the construction and study of an equilibrium Glauber-type dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space $\\Gamma$ of all locally finite subsets (configurations) in ${\\Bbb R}^d$, we fix a Gibbs measure $\\mu$ corresponding to a general pair potential $\\phi$ and activity $z>0$. We consider a Dirichlet form $ \\cal E$ on $L^2(\\Gamma,\\mu)$ which corresponds to the generator $H$ of the Glauber dynamics. We prove the existence of a Markov process $\\bf M$ on $\\Gamma$ that is properly associated with $\\cal E$. In the case of a positive potential $\\phi$ which satisfies $\\delta{:=}\\int_{{\\Bbb R}^d}(1-e^{-\\phi(x)}) z dx<1$, we also prove that the generator $H$ has a spectral gap $\\ge1-\\delta$. Furthermore, for any pure Gibbs state $\\mu$, we derive a Poincar\\'e inequality. The results about the spectral gap and the Poincar\\'e inequality are a generalization and a refinement of a recent result by L. Bertini, N. Cancrini, and F. Cesi.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-06-17T10:52:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J75",
      "60J80",
      "82C21",
      "82C22"
    ],
    "keywords": [
      "glauber dynamics",
      "poincare inequality",
      "spectral gap",
      "pure gibbs state",
      "infinite continuous particle systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......6252K"
    }
  }
}