{
  "id": "1910.10961",
  "version": "v1",
  "published": "2019-10-24T07:52:41.000Z",
  "updated": "2019-10-24T07:52:41.000Z",
  "title": "On potential theory of Markov processes with jump kernels decaying at the boundary",
  "authors": [
    "Panki Kim",
    "Renming Song",
    "Zoran Vondraček"
  ],
  "comment": "55 pages",
  "categories": [
    "math.PR",
    "math.AP",
    "math.FA"
  ],
  "abstract": "Motivated by some recent potential theoretic results on subordinate killed L\\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\\subset \\mathbb{R}^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x,y)=j(|x-y|)\\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is the L\\'evy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal L\\'evy process) in $\\mathbb{R}^d$. The main novelty is that the term $\\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\\mathcal{B}(x, y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on three parameters, $\\beta_1, \\beta_2, \\beta_3$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialize to the case of the half-space $D=\\mathbb{R}_+^d=\\{x=(\\widetilde{x},x_d):\\, x_d>0\\}$, the $\\alpha$-stable kernel $j(|x-y|)=|x-y|^{-d-\\alpha}$ and the killing function $\\kappa(x)=c x_d^{-\\alpha}$, $\\alpha\\in (0,2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(\\alpha-1)_+$, there are values of the parameters $\\beta_1, \\beta_2, \\beta_3$ and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $\\beta_1, \\beta_2, \\beta_3$ for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-24T07:52:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J45",
      "60J50",
      "60J75"
    ],
    "keywords": [
      "jump kernels decaying",
      "harnack principle fails despite",
      "potential theory",
      "markov processes",
      "boundary harnack principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}