{
  "id": "1402.6477",
  "version": "v1",
  "published": "2014-02-26T10:11:16.000Z",
  "updated": "2014-02-26T10:11:16.000Z",
  "title": "Laplacian perturbed by non-local operators",
  "authors": [
    "Jie-Ming Wang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Suppose that $d\\geq 1$ and $0<\\beta<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to the operator $\\L^b=\\Delta+S^b$, where $$ S^bf(x):=\\int_{\\R^d} \\left( f(x+z)-f(x)- \\nabla f(x) \\cdot z\\1_{\\{|z|\\leq 1\\}} \\right) \\frac{b(x, z)}{|z|^{d+\\beta}}dz$$ and $b(x, z)$ is a bounded measurable function on $\\R^d\\times \\R^d$ with $b(x, z)=b(x, -z)$ for $x, z\\in \\R^d$. We show that if for each $x\\in\\R^d, b(x, z) \\geq 0$ for a.e. $z\\in\\R^d$, then $q^b(t, x, y)$ is a strictly positive continuous function and it uniquely determines a conservative Feller process $X^b$, which has strong Feller property. Furthermore, sharp two-sided estimates on $q^b(t, x, y)$ are derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-26T10:11:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35",
      "47G20",
      "60J75",
      "47D07"
    ],
    "keywords": [
      "non-local operators",
      "strong feller property",
      "fundamental solution",
      "conservative feller process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.6477W"
    }
  }
}