{
  "id": "0706.4069",
  "version": "v2",
  "published": "2007-06-27T16:45:41.000Z",
  "updated": "2008-06-16T05:45:08.000Z",
  "title": "An effective criterion and a new example for ballistic diffusions in random environment",
  "authors": [
    "Laurent Goergen"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/07-AOP354 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2008, Vol. 36, No. 3, 1093-1133",
  "doi": "10.1214/07-AOP354",
  "categories": [
    "math.PR"
  ],
  "abstract": "In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition $(T')$, introduced by Sznitman [Ann. Probab. 29 (2001) 723--764] and by Schmitz [Ann. Inst. H. Poincar\\'{e} Probab. Statist. 42 (2006) 683--714] respectively in the discrete and continuous setting, and which implies a law of large numbers with nonvanishing limiting velocity (ballistic behavior) as well as a central limit theorem. Specifically, we show that when $d\\geq2$, $(T')$ is equivalent to an effective condition that can be checked by local inspection of the environment. When $d=1$, we prove that condition $(T')$ is merely equivalent to almost sure transience. As an application of the effective criterion, we show that when $d\\geq4$ a perturbation of Brownian motion by a random drift of size at most $\\epsilon>0$ whose projection on some direction has expectation bigger than $\\epsilon^{2-\\eta},\\eta>0$, satisfies condition $(T')$ when $\\epsilon$ is small and hence exhibits ballistic behavior. This class of diffusions contains new examples of ballistic behavior which in particular do not fulfill the condition in [Ann. Inst. H. Poincar\\'{e} Probab. Statist. 42 (2006) 683--714], (5.4) therein, related to Kalikow's condition.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-06-16T05:45:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "82D30"
    ],
    "keywords": [
      "random environment",
      "effective criterion",
      "ballistic diffusions",
      "ballistic behavior",
      "central limit theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.4069G"
    }
  }
}