{
  "id": "2209.12603",
  "version": "v1",
  "published": "2022-09-26T11:49:57.000Z",
  "updated": "2022-09-26T11:49:57.000Z",
  "title": "Green function for an asymptotically stable random walk in a half space",
  "authors": [
    "Denis Denisov",
    "Vitali Wachtel"
  ],
  "comment": "27 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider an asymptotically stable multidimensional random walk $S(n)=(S_1(n),\\ldots, S_d(n) )$. Let $\\tau_x:=\\min\\{n>0: x_{1}+S_1(n)\\le 0\\}$ be the first time the random walk $S(n)$ leaves the upper half-space. We obtain the asymptotics of $p_n(x,y):= P(x+S(n) \\in y+\\Delta, \\tau_x>n)$ as $n$ tends to infinity, where $\\Delta$ is a fixed cube. From that we obtain the local asymptotics for the Green function $G(x,y):=\\sum_n p_n(x,y)$, as $|y|$ and/or $|x|$ tend to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-26T11:49:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60G40",
      "60J45",
      "60F17"
    ],
    "keywords": [
      "asymptotically stable random walk",
      "green function",
      "half space",
      "asymptotically stable multidimensional random walk",
      "local asymptotics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}