{
  "id": "0912.1429",
  "version": "v3",
  "published": "2009-12-08T07:44:20.000Z",
  "updated": "2011-03-10T14:13:33.000Z",
  "title": "Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher",
  "authors": [
    "Atilla Yilmaz"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/10-AOP556 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2011, Vol. 39, No. 2, 471-506",
  "doi": "10.1214/10-AOP556",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on $\\mathbb{Z}^d$. There exist variational formulae for the quenched and averaged rate functions $I_q$ and $I_a$, obtained by Rosenbluth and Varadhan, respectively. $I_q$ and $I_a$ are not identically equal. However, when $d\\geq4$ and the walk satisfies the so-called (T) condition of Sznitman, they have been previously shown to be equal on an open set $\\mathcal{A}_{\\mathit {eq}}$. For every $\\xi\\in\\mathcal{A}_{\\mathit {eq}}$, we prove the existence of a positive solution to a Laplace-like equation involving $\\xi$ and the original transition kernel of the walk. We then use this solution to define a new transition kernel via the h-transform technique of Doob. This new kernel corresponds to the unique minimizer of Varadhan's variational formula at $\\xi$. It also corresponds to the unique minimizer of Rosenbluth's variational formula, provided that the latter is slightly modified.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-03-10T14:13:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large deviations",
      "random environments",
      "harmonic functions",
      "h-transform",
      "unique minimizer"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.1429Y"
    }
  }
}