{
  "id": "2403.00480",
  "version": "v1",
  "published": "2024-03-01T12:07:32.000Z",
  "updated": "2024-03-01T12:07:32.000Z",
  "title": "Markov processes with jump kernels decaying at the boundary",
  "authors": [
    "Soobin Cho",
    "Panki Kim",
    "Renming Song",
    "Zoran Vondraček"
  ],
  "comment": "123 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "The goal of this work is to develop a general theory for non-local singular operators of the type $$ L^{\\mathcal{B}}_{\\alpha}f(x)=\\lim_{\\epsilon\\to 0} \\int_{D,\\, |y-x|>\\epsilon}\\big(f(y)-f(x)\\big) \\mathcal{B}(x,y)|x-y|^{-d-\\alpha}\\,dy, $$ and $$ L f(x)=L^{\\mathcal{B}}_{\\alpha}f(x) - \\kappa(x) f(x), $$ in case $D$ is a $C^{1,1}$ open set in $\\mathbb{R}^d$, $d\\ge 2$. The function $\\mathcal{B}(x,y)$ above may vanish at the boundary of $D$, and the killing potential $\\kappa$ may be subcritical or critical. From a probabilistic point of view we study the reflected process on the closure $\\overline{D}$ with infinitesimal generator $L^{\\mathcal{B}}_{\\alpha}$, and its part process on $D$ obtained by either killing at the boundary $\\partial D$, or by killing via the killing potential $\\kappa(x)$. The general theory developed in this work (i) contains subordinate killed stable processes in $C^{1,1}$ open sets as a special case, (ii) covers the case when $\\mathcal{B}(x,y)$ is bounded between two positive constants and is well approximated by certain H\\\"older continuous functions, and (iii) extends the main results known for the half-space in $\\mathbb{R}^d$. The main results of the work are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates. Our results on the boundary Harnack principle completely cover the corresponding earlier results in the case of half-space. Our Green function estimates extend the corresponding earlier estimates in the case of half-space to bounded $C^{1, 1}$ open sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-01T12:07:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60Jxx",
      "60J45",
      "31C25",
      "35J08",
      "47G20",
      "60J46",
      "60J50",
      "60J76"
    ],
    "keywords": [
      "jump kernels decaying",
      "markov processes",
      "boundary harnack principle",
      "subordinate killed stable processes",
      "open set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 123,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}