{
  "id": "math/0409491",
  "version": "v1",
  "published": "2004-09-25T05:30:05.000Z",
  "updated": "2004-09-25T05:30:05.000Z",
  "title": "Images of the Brownian Sheet",
  "authors": [
    "Davar Khoshnevisan",
    "Yimin Xiao"
  ],
  "comment": "27 pages, submitted for publication",
  "categories": [
    "math.PR"
  ],
  "abstract": "An N-parameter Brownian sheet in R^d maps a non-random compact set F in R^N_+ to the random compact set B(F) in \\R^d. We prove two results on the image-set B(F): (1) It has positive d-dimensional Lebesgue measure if and only if F has positive (d/2)-dimensional capacity. This generalizes greatly the earlier works of J. Hawkes (1977), J.-P. Kahane (1985a; 1985b), and one of the present authors (1999). (2) If the Hausdorff dimension of F is strictly greater than (d/2), then with probability one, we can find a finite number of points \\zeta_1,...,\\zeta_m such that for any rotation matrix \\theta that leaves F in B(\\theta F), one of the \\zeta_i's is interior to B(\\theta F). In particular, B(F) has interior-points a.s. This verifies a conjecture of T. S. Mountford (1989). This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of ``sectorial local-non-determinism (LND).'' Both ideas may be of independent interest. We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-09-25T05:30:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G17",
      "28A80"
    ],
    "keywords": [
      "standard brownian motion",
      "non-random compact set",
      "n-parameter brownian sheet",
      "showcase sectorial lnd",
      "positive d-dimensional lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......9491K"
    }
  }
}