{
  "id": "1603.03902",
  "version": "v1",
  "published": "2016-03-12T12:38:50.000Z",
  "updated": "2016-03-12T12:38:50.000Z",
  "title": "The transition density of Brownian motion killed on a bounded set",
  "authors": [
    "Kohei Uchiyama"
  ],
  "comment": "21 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the transition density of a standard two-dimensional Brownian motion killed when hitting a bounded Borel set $A$. We derive the asymptotic form of the density, say $p^A_t({\\bf x},{\\bf y})$, for large times $t$ and for ${\\bf x}$ and ${\\bf y}$ in the exterior of $A$ valid uniformly under the constraint $|{\\bf x}|\\vee |{\\bf y}| =O(t)$. Within the parabolic regime $|{\\bf x}|\\vee |{\\bf y}| = O(\\sqrt t)$ in particular $p^A_t({\\bf x},{\\bf y})$ is shown to behave like $4e_A({\\bf x})e_A({\\bf y}) (\\lg t)^{-2} p_t({\\bf y}-{\\bf x})$ for large $t$, where $p_t({\\bf y}-{\\bf x})$ is the transition kernel of the Brownian motion (without killing) and $e_A$ is the Green function for the \\lq exterior of $A$' with a pole at infinity normalized so that $e_A({\\bf x}) \\sim \\lg |{\\bf x}|$. We also provide fairly accurate upper and lower bounds of $p^A_t({\\bf x},{\\bf y})$ for the case $|{\\bf x}|\\vee |{\\bf y}|>t$ as well as corresponding results for the higher dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-12T12:38:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "transition density",
      "bounded set",
      "standard two-dimensional brownian motion",
      "borel set",
      "asymptotic form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160303902U"
    }
  }
}