Weak logarithmic Sobolev inequalities and entropic convergence

Patrick Cattiaux; Ivan Gentil; Arnaud Guillin

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Weak logarithmic Sobolev inequalities and entropic convergence

Patrick Cattiaux·Ivan Gentil·Arnaud Guillin

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Abstract

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 inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.

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Patrick Cattiaux, Ivan Gentil, Arnaud Guillin. 2007-01-05. Weak logarithmic Sobolev inequalities and entropic convergence. https://arxiv.org/abs/math/0511255

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