{
  "id": "2210.11225",
  "version": "v1",
  "published": "2022-10-20T13:00:30.000Z",
  "updated": "2022-10-20T13:00:30.000Z",
  "title": "Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes",
  "authors": [
    "Kyung-Youn Kim",
    "Lidan Wang"
  ],
  "comment": "34 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $Z=(Z^{1}, \\ldots, Z^{d})$ be the d-dimensional L\\'evy {process} where {$Z^i$'s} are independent 1-dimensional L\\'evy {processes} with identical jumping kernel $ \\nu^1(r) =r^{-1}\\phi(r)^{-1}$. Here $\\phi$ is {an} increasing function with weakly scaling condition of order $\\underline \\alpha, \\overline \\alpha\\in (0, 2)$. We consider a symmetric function $J(x,y)$ comparable to \\begin{align*} \\begin{cases} \\nu^1(|x^i - y^i|)\\qquad&\\text{ if $x^i \\ne y^i$ for some $i$ and $x^j = y^j$ for all $j \\ne i$}\\\\ 0\\qquad&\\text{ if $x^i \\ne y^i$ for more than one index $i$}. \\end{cases} \\end{align*} Corresponding to the jumping kernel $J$, there exists an anisotropic Markov process $X$, see \\cite{KW22}. In this article, we establish sharp two-sided Dirichlet heat kernel estimates for $X$ in $C^{1,1}$ open set, under certain regularity conditions. As an application of the main results, we derive the Green function estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-20T13:00:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B25",
      "60J50"
    ],
    "keywords": [
      "anisotropic markov process",
      "large class",
      "two-sided dirichlet heat kernel estimates",
      "sharp two-sided dirichlet heat kernel"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}