{
  "id": "math/0407318",
  "version": "v1",
  "published": "2004-07-19T01:49:17.000Z",
  "updated": "2004-07-19T01:49:17.000Z",
  "title": "α-Continuity Properties of Stable Processes",
  "authors": [
    "R. D. DeBlassie",
    "Pedro J. Mendez-Hernandez"
  ],
  "comment": "22 pages",
  "categories": [
    "math.PR",
    "math.SP"
  ],
  "abstract": "Let $D$ be a domain of finite Lebesgue measure in $\\bR^d$ and let $X^D_t$ be the symmetric $\\alpha$-stable process killed upon exiting $D$. Each element of the set $\\{\\lambda_i^\\alpha\\}_{i=1}^\\infty$ of eigenvalues associated to $X^D_t$, regarded as a function of $\\alpha\\in(0,2)$, is right continuous. In addition, if $D$ is Lipschitz and bounded, then each $ \\lambda_i^\\alpha$ is continuous in $\\alpha$ and the set of associated eigenfunctions is precompact. We also prove that if $D$ is a domain of finite Lebesgue measure, then for all $0<\\alpha<\\beta\\leq 2$ and $i\\geq 1$, \\[\\lambda_i^\\alpha \\leq [ \\lambda^\\beta_i]^{\\alpha/\\beta}.\\] Previously, this bound had been known only for $\\beta=2$ and $\\alpha$ rational.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-07-19T01:49:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J45",
      "26A33"
    ],
    "keywords": [
      "stable processes",
      "finite lebesgue measure",
      "properties",
      "eigenvalues",
      "associated eigenfunctions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......7318D"
    }
  }
}