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, etc.). 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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Cattiaux, P., Gentil, I. & Guillin, A. Weak logarithmic Sobolev inequalities and entropic convergence. Probab. Theory Relat. Fields 139, 563–603 (2007). https://doi.org/10.1007/s00440-007-0054-5
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DOI: https://doi.org/10.1007/s00440-007-0054-5
Keywords
- Logarithmic Sobolev inequalities
- Concentration inequalities
- Entropy
Mathematics Subject Classification (2000)
- 26D10
- 60E15