# Some remarks on weighted logarithmic Sobolev inequality

@article{Cattiaux2010SomeRO,
title={Some remarks on weighted logarithmic Sobolev inequality},
author={Patrick Cattiaux and Arnaud Guillin and Liming Wu},
journal={arXiv: Probability},
year={2010}
}
• Published 21 May 2010
• Mathematics
• arXiv: Probability
We give here a simple proof of weighted logarithmic Sobolev inequality, for example for Cauchy type measures, with optimal weight, sharpening results of Bobkov-Ledoux. Some consequences are also discussed.

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## References

SHOWING 1-10 OF 41 REFERENCES

### Modified logarithmic Sobolev inequalities in null curvature

• Mathematics
• 2005
We present a new logarithmic Sobolev inequality adapted to a log-concave measure on $\dR$ between the exponential and the Gaussian measure.

### Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities

• Mathematics
• 1999
Abstract We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a

### Weak logarithmic Sobolev inequalities and entropic convergence

• Mathematics
• 2005
In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner

### A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality

• Mathematics
• 2008
We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand’s transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient

### Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution

• Mathematics
• 1997
Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every

### A Pathwise Approach of Some Classical Inequalities

The aim of this pedagogical paper is to show how some renowned inequalities may be obtained via a simple argument: entropy projection from the path space onto finite-dimensional coordinates spaces.

### A simple proof of the Poincaré inequality for a large class of probability measures

• Mathematics
• 2008
Abstract. We give a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on

### Mass Transport and Variants of the Logarithmic Sobolev Inequality

• Mathematics
• 2007
We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are

### METRIC HOPF-LAX FORMULA WITH SEMICONTINUOUS DATA

In this paper we study a metric Hopf-Lax formula looking in particular at the Carnot-Caratheodory case. We generalize many properties of the classical euclidean Hopf-Lax formula and we use it in