{
  "id": "1010.2987",
  "version": "v1",
  "published": "2010-10-14T17:33:58.000Z",
  "updated": "2010-10-14T17:33:58.000Z",
  "title": "Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension",
  "authors": [
    "Yuval Peres",
    "Perla Sousi"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "By the Cameron--Martin theorem, if a function $f$ is in the Dirichlet space $D$, then $B+f$ has the same a.s. properties as standard Brownian motion, $B$. In this paper we examine properties of $B+f$ when $f \\notin D$. We start by establishing a general 0-1 law, which in particular implies that for any fixed $f$, the Hausdorff dimension of the image and the graph of $B+f$ are constants a.s. (This 0-1 law applies to any L\\'evy process.) Then we show that if the function $f$ is H\\\"older$(1/2)$, then $B+f$ is intersection equivalent to $B$. Moreover, $B+f$ has double points a.s. in dimensions $d\\le 3$, while in $d\\ge 4$ it does not. We also give examples of functions which are H\\\"older with exponent less than $1/2$, that yield double points in dimensions greater than 4. Finally, we show that for $d \\ge 2$, the Hausdorff dimension of the image of $B+f$ is a.s. at least the maximum of 2 and the dimension of the image of $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-14T17:33:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hausdorff dimension",
      "hitting probabilities",
      "variable drift",
      "standard brownian motion",
      "double points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.2987P"
    }
  }
}