{
  "id": "1709.02836",
  "version": "v1",
  "published": "2017-09-08T19:12:43.000Z",
  "updated": "2017-09-08T19:12:43.000Z",
  "title": "Heat kernel estimates for non-symmetric stable-like processes",
  "authors": [
    "Peng Jin"
  ],
  "comment": "37 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $d\\ge1$ and $0<\\alpha<2$. Consider the integro-differential operator \\[ \\mathcal{L}f(x) =\\int_{\\mathbb{R}^{d}\\backslash\\{0\\}}\\left[f(x+h)-f(x)-\\chi_{\\alpha}(h)\\nabla f(x)\\cdot h\\right]\\frac{n(x,h)}{|h|^{d+\\alpha}}\\mathrm{d}h+\\mathbf{1}_{\\alpha>1}b(x)\\cdot\\nabla f(x), \\] where $\\chi_{\\alpha}(h):=\\mathbf{1}_{\\alpha>1}+\\mathbf{1}_{\\alpha=1}\\mathbf{1}_{\\{|h|\\le1\\}}$, $b:\\mathbb{R}^{d}\\to\\mathbb{R}^{d}$ is bounded measurable, and $n:\\mathbb{R}^{d}\\times\\mathbb{R}^{d}\\to\\mathbb{R}$ is measurable and bounded above and below respectively by two positive constants. Further, we assume that $n(x,h)$ is H\\\"older continuous in $x$, uniformly with respect to $h\\in\\mathbb{R}^{d}$. In the case $\\alpha=1,$ we assume additionally $\\int_{\\partial B_{r}}n(x,h)h\\mathrm{d}S_{r}(h)=0$, $\\forall r \\in (0,\\infty)$, where $\\mathrm{d}S_{r}$ is the surface measure on $\\partial B_{r}$, the boundary of the ball with radius $r$ and center $0$. In this paper, we establish two-sided estimates for the heat kernel of the Markov process associated with the operator $\\mathcal{L}$. This extends a recent result of Z.-Q. Chen and X. Zhang.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-08T19:12:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35",
      "47G20",
      "60J75"
    ],
    "keywords": [
      "heat kernel estimates",
      "non-symmetric stable-like processes",
      "integro-differential operator",
      "surface measure",
      "markov process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}