{
  "id": "1103.4622",
  "version": "v3",
  "published": "2011-03-23T20:30:43.000Z",
  "updated": "2012-01-30T20:20:56.000Z",
  "title": "Spectral condition, hitting times and Nash inequality",
  "authors": [
    "Eva Loecherbach",
    "Dasha Loukianova",
    "Oleg Loukianov"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ be a $\\mu$-symmetric Hunt process on a LCCB space E. For an open set G $\\subseteq$ E, let $\\tau_G$ be the exit time of $X$ from G and $A^G$ be the generator of the process killed when it leaves G. Let $r:[0,\\infty[\\to[0,\\infty[$ and $R (t) = \\int_0^t r(s) ds$. We give necessary and sufficient conditions for $\\E_{\\mu} R (\\tau_G)<\\infty$ in terms of the behavior near the origin of the spectral measure of $-A^G.$ When $r(t)=t^l$, $l>0$, by means of this condition we derive the Nash inequality for the killed process. In the case of one-dimensional diffusions, this permits to show that the existence of moments of order $l$ for $\\tau_G$ implies the Nash inequality of order $p=\\frac{l+2}{l+1}$ for the whole process. The associated rate of convergence of the semi-group in $L^2(\\mu)$ is bounded by $t^{-(l+1)}$. For diffusions in dimension greater than one, we obtain the Nash inequality of the same order under an additional non-degeneracy condition (local Poincar\\'e inequality). Finally, we show for general Hunt processes that the Nash inequality giving rise to a convergence rate of order $t^{-(l+1)}$ of the semi-group, implies the existence of moments of order $l+1 -\\epsilon$ for $\\tau_G$, for all $ \\epsilon>0$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-01-30T20:20:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J25",
      "60J35",
      "60J60"
    ],
    "keywords": [
      "spectral condition",
      "hitting times",
      "general hunt processes",
      "nash inequality giving rise",
      "local poincare inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.4622L"
    }
  }
}