{
  "id": "0707.2908",
  "version": "v3",
  "published": "2007-07-19T14:12:01.000Z",
  "updated": "2012-01-04T12:33:35.000Z",
  "title": "Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence",
  "authors": [
    "Sébastien Chambeu",
    "Aline Kurtzmann"
  ],
  "comment": "Published in at http://dx.doi.org/10.3150/10-BEJ310 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)",
  "journal": "Bernoulli 2011, Vol. 17, No. 4, 1248-1267",
  "doi": "10.3150/10-BEJ310",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "This paper deals with some self-interacting diffusions $(X_t,t\\geq 0)$ living on $\\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: \\[\\mathrm{d}X_t=\\mathrm{d}B_t-g(t)\\nabla V(X_t-\\bar{\\mu}_t)\\,\\mathrm{d}t,\\] where $\\bar{\\mu}_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behaviour of $X$ and prove that it is strongly related to $g$. Actually, we show that $X$ is ergodic (in the limit quotient sense) if and only if $\\bar{\\mu}_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-01-04T12:33:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sure convergence",
      "ergodic behaviour",
      "self-interacting diffusions",
      "stochastic differential equations",
      "limit quotient sense"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.2908C"
    }
  }
}