{
  "id": "2110.11653",
  "version": "v2",
  "published": "2021-10-22T08:29:06.000Z",
  "updated": "2022-03-20T17:35:06.000Z",
  "title": "Potential theory of Dirichlet forms degenerate at the boundary: the case of no killing potential",
  "authors": [
    "Panki Kim",
    "Renming Song",
    "Zoran Vondraček"
  ],
  "comment": "A small gap in the proof of Lemma 4.3 fixed",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we consider the Dirichlet form on the half-space $\\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-\\alpha}\\mathcal{B}(x,y)$, where $\\mathcal{B}(x,y)$ can be degenerate at the boundary. Unlike our previous works [6,7] where we imposed critical killing, here we assume that the killing potential is identically zero. In case $\\alpha\\in (1,2)$ we first show that the corresponding Hunt process has finite lifetime and dies at the boundary. Then, as our main contribution, we prove the boundary Harnack principle and establish sharp two-sided Green function estimates. Our results cover the case of the censored $\\alpha$-stable process, $\\alpha\\in (1,2)$, in the half-space studied in [2].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-20T17:35:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J45",
      "60J50",
      "60J76"
    ],
    "keywords": [
      "dirichlet forms degenerate",
      "killing potential",
      "potential theory",
      "sharp two-sided green function estimates",
      "establish sharp two-sided green function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}