{
  "id": "math/0611565",
  "version": "v1",
  "published": "2006-11-19T06:26:04.000Z",
  "updated": "2006-11-19T06:26:04.000Z",
  "title": "On the Estimates of the Density of Feynman-Kac Semigroups of $α$-Stable-like Processes",
  "authors": [
    "Chunlin Wang"
  ],
  "comment": "27 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Suppose that $\\alpha \\in (0,2)$ and that $X$ is an $\\alpha$-stable-like process on $\\R^d$. Let $F$ be a function on $\\R^d$ belonging to the class $\\bf{J_{d,\\alpha}}$ (see Introduction) and $A_{t}^{F}$ be $\\sum_{s \\le t}F(X_{s-},X_{s}), t> 0$, a discontinuous additive functional of $X$. With neither $F$ nor $X$ being symmetric, under certain conditions, we show that the Feynman-Kac semigroup $\\{S_{t}^{F}:t \\ge 0\\}$ defined by $$ S_{t}^{F}f(x)=\\mathbb{E}_{x}(e^{-A_{t}^{F}}f(X_{t}))$$ has a density $q$ and that there exist positive constants $C_1,C_2,C_3$ and $C_4$ such that $$C_{1}e^{-C_{2}t}t^{-\\frac{d}{\\alpha}}(1 \\wedge \\frac{t^{\\frac{1}{\\alpha}}}{|x-y|})^{d+\\alpha} \\leq q(t,x,y) \\leq C_{3}e^{C_{4}t}t^{-\\frac{d}{\\alpha}}(1 \\wedge \\frac{t^{\\frac{1}{\\alpha}}}{|x-y|})^{d+\\alpha}$$ for all $(t,x,y)\\in (0,\\infty) \\times \\R^d \\times \\R^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-11-19T06:26:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J45",
      "60J35",
      "60J55",
      "60J75",
      "60G51",
      "60G52"
    ],
    "keywords": [
      "feynman-kac semigroup",
      "stable-like processes",
      "introduction",
      "discontinuous additive functional",
      "conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11565W"
    }
  }
}