{
  "id": "1002.1121",
  "version": "v1",
  "published": "2010-02-05T04:30:59.000Z",
  "updated": "2010-02-05T04:30:59.000Z",
  "title": "Heat Kernel Estimate for $Δ+Δ^{α/2}$ in $C^{1,1}$ open sets",
  "authors": [
    "Zhen-Qing Chen",
    "Panki Kim",
    "Renming Song"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We consider a family of pseudo differential operators $\\{\\Delta+ a^\\alpha \\Delta^{\\alpha/2}; a\\in (0, 1]\\}$ on $\\bR^d$ for every $d\\geq 1$ that evolves continuously from $\\Delta$ to $\\Delta + \\Delta^{\\alpha/2}$, where $\\alpha \\in (0, 2)$. It gives rise to a family of L\\'evy processes $\\{X^a, a\\in (0, 1]\\}$ in $\\bR^d$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\\alpha$-stable process with weight $a$. We establish sharp two-sided estimates for the heat kernel of $\\Delta + a^{\\alpha} \\Delta^{\\alpha/2}$ with zero exterior condition in a family of open subsets, including bounded $C^{1, 1}$ (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric $\\alpha$-stable process with weight $a$ in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in $a$ in the sense that the constants in the estimates are independent of $a\\in (0, 1]$ so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking $a\\to 0$. Integrating the heat kernel estimates in time $t$, we recover the two-sided sharp uniform Green function estimates of $X^a$ in bounded $C^{1,1}$ open sets in $\\bR^d$, which were recently established in \\cite{CKSV2} by using a completely different approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-02-05T04:30:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35",
      "47G20",
      "60J75",
      "47D07"
    ],
    "keywords": [
      "open sets",
      "brownian motion",
      "sharp two-sided estimates",
      "sharp uniform green function estimates",
      "dirichlet heat kernel estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.1121C"
    }
  }
}