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April, 2007 Modified logarithmic Sobolev inequalities in null curvature
Ivan Gentil, Arnaud Guillin, Laurent Miclo
Rev. Mat. Iberoamericana 23(1): 235-258 (April, 2007).

Abstract

We present a new logarithmic Sobolev inequality adapted to a log-concave measure on $\mathbb{R}$ between the exponential and the Gaussian measure. More precisely, assume that $\Phi$ is a symmetric convex function on $\mathbb{R}$ satisfying $(1+\varepsilon)\Phi(x)\leq {x}\Phi'(x)\leq(2-\varepsilon)\Phi(x)$ for $x\geq 0$ large enough and with $\varepsilon\in ]0,1/2]$. We prove that the probability measure on $\mathbb{R}$ $\mu_\Phi(dx)=e^{-\Phi(x)}/Z_\Phi dx$ satisfies a modified and adapted logarithmic Sobolev inequality: there exist three constants $A,B,C>0$ such that for all smooth functions $f>0$, $$ \mathbf{Ent}_{\mu_\Phi}{\left(f^2\right)}\leq A\int H_{\Phi}\left(\frac{f'}{f}\right)f^2d\mu_\Phi, $$ with $$ H_{\Phi}(x)= \left\{ \begin{array}{l} x^2 \text{ if }\left|x\right|< C,\\ \Phi^*\left(Bx\right) \text{ if }\left|x\right|\geq C, \end{array} \right. $$ where $\Phi^*$ is the Legendre-Fenchel transform of $\Phi$.

Citation

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Ivan Gentil. Arnaud Guillin. Laurent Miclo. "Modified logarithmic Sobolev inequalities in null curvature." Rev. Mat. Iberoamericana 23 (1) 235 - 258, April, 2007.

Information

Published: April, 2007
First available in Project Euclid: 1 June 2007

zbMATH: 1123.26022
MathSciNet: MR2351133

Subjects:
Primary: 26D99, 39B72, 60E15

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.23 • No. 1 • April, 2007
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