{
  "id": "1312.7594",
  "version": "v2",
  "published": "2013-12-29T21:56:36.000Z",
  "updated": "2016-09-29T07:39:03.000Z",
  "title": "Perturbation by non-local operators",
  "authors": [
    "Zhen-Qing Chen",
    "Jie-Ming Wang"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "Suppose that $d\\ge 1$ and $0<\\beta<\\alpha<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to a class of (possibly nonsymmetric) non-local operators $L^b=\\Delta^{\\alpha/2}+S^b$, where $$ S^bf(x):=A(d, -\\beta) \\int_{R^d} ( f(x+z)-f(x)- \\nabla f(x) \\cdot z 1_{\\{|z|\\leq 1\\}} ) \\frac{b(x, z)}{|z|^{d+\\beta}}dz $$ and $b(x, z)$ is a bounded measurable function on $R^d\\times R^d$ with $b(x, z)=b(x, -z)$ for $x, z\\in R^d$. Here $A(d, -\\beta)$ is a normalizing constant so that $S^b=\\Delta^{\\beta/2}$ when $b(x, z)\\equiv 1$. We show that if $b(x, z) \\geq -\\frac{{\\cal A}(d, -\\alpha)}{A(d, -\\beta)}\\, |z|^{\\beta -\\alpha}$, 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. The Feller process $X^b$ is the unique solution to the martingale problem of $(L^b, {\\cal S} (R^d))$, where ${\\cal S}(R^d)$ denotes the space of tempered functions on $R^d$. Furthermore, sharp two-sided estimates on $q^b(t, x, y)$ are derived. In stark contrast with the gradient perturbations, these estimates exhibit different behaviors for different types of $b(x, z)$. The model considered in this paper contains the following as a special case. Let $Y$ and $Z$ be (rotationally) symmetric $\\alpha$-stable process and symmetric $\\beta$-stable processes on $R^d$, respectively, that are independent to each other. Solution to stochastic differential equations $dX_t=dY_t + c(X_{t-})dZ_t$ has infinitesimal generator $L^b$ with $b(x, z)=| c(x)|^\\beta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-29T21:56:36.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-09-29T07:39:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35",
      "47G20",
      "60J75",
      "47D07"
    ],
    "keywords": [
      "non-local operators",
      "stochastic differential equations",
      "strong feller property",
      "special case",
      "paper contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.7594C"
    }
  }
}