{
  "id": "1208.5112",
  "version": "v3",
  "published": "2012-08-25T07:49:05.000Z",
  "updated": "2013-01-30T09:09:47.000Z",
  "title": "Green function estimates for subordinate Brownian motions : stable and beyond",
  "authors": [
    "Panki Kim",
    "Ante Mimica"
  ],
  "comment": "We have weaken the condition (A5). References are updated",
  "categories": [
    "math.PR"
  ],
  "abstract": "A subordinate Brownian motion $X$ is a L\\'evy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $\\phi$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded $C^{1,1}$ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{1,1}$ open set with explicit decay rate. Unlike {KSV2, KSV4}, our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent $$\\phi(\\lambda)=\\log(1+\\lambda^{\\alpha/2}) (0<\\alpha\\leq 2, d > \\alpha)$$ and $$\\phi(\\lambda)=\\log(1+(\\lambda+m^{\\alpha/2})^{2/\\alpha}-m) (0<\\alpha<2,\\, m>0, d >2) .$$",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-01-30T09:09:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J45",
      "60J75",
      "60G51"
    ],
    "keywords": [
      "subordinate brownian motion",
      "green function estimates",
      "open set",
      "scale invariant boundary harnack inequality",
      "results cover geometric stable processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.5112K"
    }
  }
}