{
  "id": "math/0310306",
  "version": "v2",
  "published": "2003-10-20T08:26:36.000Z",
  "updated": "2004-10-23T04:36:19.000Z",
  "title": "Diffusion in random environment and the renewal theorem",
  "authors": [
    "Dimitrios Cheliotis"
  ],
  "comment": "18 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "According to a theorem of S. Schumacher and T. Brox, for a diffusion $X$ in a Brownian environment it holds that $(X_t-b_{\\log t})/\\log^2t\\to 0 $ in probability, as $t\\to\\infty$, where $b_{\\cdot}$ is a stochastic process having an explicit description and depending only on the environment. We compute the distribution of the number of sign changes for $b$ on an interval $[1,x]$ and study some of the consequences of the computation; in particular we get the probability of $b$ keeping the same sign on that interval. These results have been announced in 1999 in a non-rigorous paper by P. Le Doussal, C. Monthus, and D. Fisher and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. Finally, we comment on the information these results give about the behavior of the diffusion.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-10-23T04:36:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G52"
    ],
    "keywords": [
      "random environment",
      "renewal theorem",
      "brownian environment",
      "renormalization group analysis",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....10306C"
    }
  }
}