{
  "id": "1408.2883",
  "version": "v1",
  "published": "2014-08-13T00:10:26.000Z",
  "updated": "2014-08-13T00:10:26.000Z",
  "title": "Effective dimension of points visited by Brownian motion",
  "authors": [
    "Bjørn Kjos-Hanssen",
    "Anil Nerode"
  ],
  "comment": "The conference version was published as: The law of the iterated logarithm for algorithmically random paths of Brownian motion, Logical Foundations of Computer Science, Lecture Notes in Computer Science 4514 (2007), 310--317",
  "journal": "Theoretical Computer Science 410 (2009), no. 4-5, 347--354",
  "doi": "10.1016/j.tcs.2008.09.045",
  "categories": [
    "math.LO"
  ],
  "abstract": "We consider the individual points on a Martin-L\\\"of random path of Brownian motion. We show (1) that Khintchine's law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension $<1$. The proof of (1) shows that for almost all times $t$, the path $f$ is Martin-L\\\"of random relative to $t$ and so the effective dimension of $(t,f(t))$ is 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-13T00:10:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "effective dimension",
      "brownian motion",
      "random path",
      "trivial example"
    ],
    "tags": [
      "conference paper",
      "journal article",
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2883K"
    }
  }
}