{
  "id": "math/0702178",
  "version": "v2",
  "published": "2007-02-07T11:23:21.000Z",
  "updated": "2007-08-20T12:44:48.000Z",
  "title": "Diffusion approximation for equilibrium Kawasaki dynamics in continuum",
  "authors": [
    "Y. G. Kondratiev",
    "O. V. Kutoviy",
    "E. W. Lytvynov"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in $\\mathbb R^d$ which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure $\\mu$ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time. Under weak assumptions on the potential of pair interaction, $\\phi$, (in particular, admitting a singularity of $\\phi$ at zero), we prove that, on a set of smooth local functions, the generator of the scaled dynamics converges to the generator of the gradient stochastic dynamics. If the set on which the generators converge is a core for the diffusion generator, the latter result implies the weak convergence of finite-dimensional distributions of the corresponding equilibrium processes. In particular, if the potential $\\phi$ is from $C_{\\mathrm b}^3(\\mathbb R^d)$ and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [Choi {\\it et al.}, J. Math. Phys. 39 (1998) 6509--6536].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-08-20T12:44:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F99",
      "60J60",
      "60J75",
      "60K35"
    ],
    "keywords": [
      "equilibrium kawasaki dynamics",
      "diffusion approximation",
      "gradient stochastic dynamics",
      "smooth local functions",
      "finite-dimensional distributions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......2178K"
    }
  }
}