{
  "id": "1608.06348",
  "version": "v1",
  "published": "2016-08-23T00:36:35.000Z",
  "updated": "2016-08-23T00:36:35.000Z",
  "title": "Asymptotic behaviour of a random walk killed on a finite set",
  "authors": [
    "Kohei Uchiyama"
  ],
  "comment": "16 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study asymptotic behavior, for large time $n$, of the transition probability of a two-dimensional random walk killed when entering into a non-empty finite subset $A$. We show that it behaves like $4 \\tilde u_A(x) \\tilde u_{-A}(-y) (\\lg n)^{-2} p^n(y- x)$ for large $n$, uniformly in the parabolic regime $|x|\\vee |y| =O(\\sqrt n)$, where $p^n(y-x)$ is the transition kernel of the random walk (without killing) and $\\tilde u_A$ is the unique harmonic function in the 'exterior of $A$' satisfying the boundary condition $\\tilde u_A(x) \\sim \\lg |x|$ at infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-23T00:36:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite set",
      "asymptotic behaviour",
      "unique harmonic function",
      "non-empty finite subset",
      "study asymptotic behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}