{
  "id": "math/0503585",
  "version": "v1",
  "published": "2005-03-25T08:58:54.000Z",
  "updated": "2005-03-25T08:58:54.000Z",
  "title": "Modified logarithmic Sobolev inequalities in null curvature",
  "authors": [
    "Ivan Gentil",
    "Arnaud Guillin",
    "Laurent Miclo"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We present a logarithmic Sobolev inequality adapted to a log-concave measure. Assume that $\\Phi$ is a symmetric convex function on $\\dR$ satisfying $(1+\\e)\\Phi(x)\\leq {x}\\Phi'(x)\\leq(2-\\e)\\Phi(x)$ for $x\\geq0$ large enough and with $\\e\\in]0,1/2]$. We prove that the probability measure on $\\dR$ $\\mu_\\Phi(dx)=e^{-\\Phi(x)}/Z_\\Phi dx$ satisfies a modified and adapted logarithmic Sobolev inequality : there exist three constant $A,B,D>0$ such that for all smooth $f>0$, \\begin{equation*} \\ent{\\mu_\\Phi}{f^2}\\leq A\\int H_{\\Phi}\\PAR{{\\frac{f'}{f}}}f^2d\\mu_\\Phi, \\text{with} H_{\\Phi}(x)= {\\begin{array}{rl} \\Phi^*\\PAR{Bx} &\\text{if }\\ABS{x}\\geq D, x^2 &\\text{if}\\ABS{x}\\leq D. \\end{array} . \\end{equation*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-03-25T08:58:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modified logarithmic sobolev inequalities",
      "null curvature",
      "adapted logarithmic sobolev inequality",
      "symmetric convex function",
      "log-concave measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......3585G"
    }
  }
}