{
  "id": "1501.04595",
  "version": "v1",
  "published": "2015-01-19T19:26:16.000Z",
  "updated": "2015-01-19T19:26:16.000Z",
  "title": "Asymptotics for the heat kernel in multicone domains",
  "authors": [
    "Pierre Collet",
    "Mauricio Duarte",
    "Servet Martinez",
    "Arturo Prat-Waldron",
    "Jaime San Martin"
  ],
  "comment": "Submitted to The Annals of Probability",
  "categories": [
    "math.PR"
  ],
  "abstract": "A multi cone domain $\\Omega \\subseteq \\mathbb{R}^n$ is a domain that resembles a finite collection of cones far away from the origin. We study the rate of decay in time of the heat kernel $p(t,x,y)$ of a Brownian motion killed upon exiting $\\Omega$, using both probabilistic and analytical techniques. We find that the decay is polynomial and we characterize $\\lim_{t\\to\\infty} t^{1+\\alpha}p(t,x,y)$ in terms of the Martin boundary of $\\Omega$ at infinity, where $\\alpha>0$ depends on the geometry of $\\Omega$. We next derive an analogous result for $t^{\\kappa/2}\\mathbb{P}_x(T >t)$, with $\\kappa = 1+\\alpha - n/2$, where $T$ is the exit time form $\\Omega$. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-19T19:26:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65",
      "35K08",
      "35B40"
    ],
    "keywords": [
      "heat kernel",
      "multicone domains",
      "asymptotics",
      "cones far away",
      "exit time form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150104595C"
    }
  }
}