JournalsrmiVol. 23, No. 1pp. 235–258

Modified logarithmic Sobolev inequalities in null curvature

  • Ivan Gentil

    Université Claude Bernard Lyon 1, Villeurbanne, France
  • Arnaud Guillin

    Université de Provence, Marseille, France
  • Laurent Miclo

    Université de Provence, Marseille, France
Modified logarithmic Sobolev inequalities in null curvature cover
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Abstract

We present a new logarithmic Sobolev inequality adapted to a log-concave measure on R\mathbb{R} between the exponential and the Gaussian measure. More precisely, assume that Φ\Phi is a symmetric convex function on R\mathbb{R} satisfying (1+ε)Φ(x)xΦ(x)(2ε)Φ(x)(1+\varepsilon)\Phi(x)\leq {x}\Phi'(x)\leq(2-\varepsilon)\Phi(x) for x0x\geq 0 large enough and with ε]0,1/2]\varepsilon\in ]0,1/2]. We prove that the probability measure on R\mathbb{R} μΦ(dx)=eΦ(x)/ZΦdx\mu_\Phi(dx)=e^{-\Phi(x)}/Z_\Phi dx satisfies a modified and adapted logarithmic Sobolev inequality: there exist three constants A,B,C>0A,B,C>0 such that for all smooth functions f>0f>0,

EntμΦ(f2)AHΦ(ff)f2dμΦ,\mathbf{Ent}_{\mu_\Phi}{\left(f^2\right)}\leq A\int H_{\Phi}\left(\frac{f'}{f}\right)f^2d\mu_\Phi,

with

HΦ(x)={x2 if x<C,Φ(Bx) if xC,H_{\Phi}(x)= \left\{ \begin{array}{l} x^2 \text{ if }\left|x\right|< C,\\ \Phi^*\left(Bx\right) \text{ if }\left|x\right|\geq C, \end{array} \right.

where Φ\Phi^* is the Legendre-Fenchel transform of Φ\Phi.

Cite this article

Ivan Gentil, Arnaud Guillin, Laurent Miclo, Modified logarithmic Sobolev inequalities in null curvature. Rev. Mat. Iberoam. 23 (2007), no. 1, pp. 235–258

DOI 10.4171/RMI/493