  In this paper we introduce and study a weakened form of logarithmic Sobolev
inequalities in connection with various others functional inequalities (weak
Poincar\'{e} 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 Poincar\'{e} inequality
can not be used for deriving entropic convergence whence weak logarithmic
Sobolev inequality ensures the result.

Probability