{
  "id": "0902.2785",
  "version": "v1",
  "published": "2009-02-16T21:07:36.000Z",
  "updated": "2009-02-16T21:07:36.000Z",
  "title": "Random Walks in the Quarter Plane Absorbed at the Boundary : Exact and Asymptotic",
  "authors": [
    "Kilian Raschel"
  ],
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "Nearest neighbor random walks in the quarter plane that are absorbed when reaching the boundary are studied. The cases of positive and zero drift are considered. Absorption probabilities at a given time and at a given site are made explicit. The following asymptotics for these random walks starting from a given point $(n_0,m_0)$ are computed : that of probabilities of being absorbed at a given site $(i,0)$ [resp. $(0,j)$] as $i\\to \\infty$ [resp. $j \\to \\infty$], that of the distribution's tail of absorption time at x-axis [resp. y-axis], that of the Green functions at site $(i,j)$ when $i,j\\to \\infty$ and $j/i \\to \\tan \\gamma$ for $\\gamma \\in [0, \\pi/2]$. These results give the Martin boundary of the process and in particular the suitable Doob $h$-transform in order to condition the process never to reach the boundary. They also show that this $h$-transformed process is equal in distribution to the limit as $n\\to \\infty$ of the process conditioned by not being absorbed at time $n$. The main tool used here is complex analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-16T21:07:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60G40",
      "31C35",
      "30E20",
      "30E25"
    ],
    "keywords": [
      "quarter plane",
      "asymptotic",
      "nearest neighbor random walks",
      "probabilities",
      "complex analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.2785R"
    }
  }
}