{
  "id": "1402.4660",
  "version": "v1",
  "published": "2014-02-19T13:47:10.000Z",
  "updated": "2014-02-19T13:47:10.000Z",
  "title": "Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in $C^{1,η}$ open sets",
  "authors": [
    "Kyung-Youn Kim",
    "Panki Kim"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in $C^{1,\\eta}$ open sets. The processes are symmetric pure jump Markov processes with jumping intensity $\\kappa(x,y) \\psi_1 (|x-y|)^{-1} |x-y|^{-d-\\alpha}$, where $\\alpha \\in (0,2)$. Here, $\\psi_1$ is an increasing function on $[ 0, \\infty )$, with $\\psi_1(r)=1$ on $0<r \\le 1$ and $c_1e^{c_2r^{\\beta}} \\le \\psi_1(r) \\le c_3 e^{c_4r^{\\beta}}$ on $r>1$ for $\\beta \\in [0,\\infty]$, and $ \\kappa( x, y)$ is a symmetric function confined between two positive constants, with $|\\kappa(x,y)-\\kappa(x,x)|\\leq c_5|x-y|^{\\rho}$ for $|x-y|<1$ and $\\rho>\\alpha/2$. We establish two-sided estimates for the transition densities of such processes in $C^{1,\\eta}$ open sets when $\\eta \\in (\\alpha/2, 1]$. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in $C^{1,\\eta}$ open sets when $\\eta \\in (\\alpha/2, 1]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-19T13:47:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35",
      "47G20",
      "60J75",
      "47D07"
    ],
    "keywords": [
      "open sets",
      "symmetric markov processes",
      "transition densities",
      "two-sided estimates",
      "dirichlet heat kernel estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.4660K"
    }
  }
}