{
  "id": "1410.8240",
  "version": "v1",
  "published": "2014-10-30T03:23:30.000Z",
  "updated": "2014-10-30T03:23:30.000Z",
  "title": "Heat kernel estimates for $Δ+Δ^{α/2}$ under gradient perturbation",
  "authors": [
    "Zhen-Qing Chen",
    "Eryan Hu"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "For $d \\ge 2$, $\\alpha \\in (0,2)$ and $M > 0$, we consider the gradient perturbation of a family of nonlocal operators $\\{\\Delta+a^\\alpha\\Delta^{\\alpha/2}, a\\in (0,M]\\}$. We establish the existence and uniqueness of the fundamental solution $p(t, x, y)$ for \\begin{equation*} \\mathcal{L}^{a,b} = \\Delta+a^\\alpha\\Delta^{\\alpha/2} + b\\cdot \\nabla, \\end{equation*} where $b$ is in Kato class $\\mathbb{K}_{d,1}$ on $\\mathbb{R}^d$. We show that $p(t, x, y)$ is jointly continuous and derive its sharp two-sided estimates. The kernel $p(t, x, y)$ determines a conservative Feller process $X$. We further show that the law of $X$ is the unique solution of the martingale problem for $(\\mathcal{L}^{a,b}, C^\\infty_c (\\mathbb{R}^d)$ and $X$ can be represented as $$ X_t = X_0 + Z^a_t + \\int_0^t b(X_s) ds, \\qquad t\\geq 0, $$ where $Z^a_t= B_t +aY_t$ for a Brownian motion $B$ and an independent isotropic $\\alpha$-stable process $Y$. Moreover, we prove that the above SDE has a unique weak solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-30T03:23:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35",
      "60H10",
      "35K08",
      "47G20",
      "47D07"
    ],
    "keywords": [
      "heat kernel estimates",
      "gradient perturbation",
      "unique weak solution",
      "independent isotropic",
      "brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.8240C"
    }
  }
}