Weak logarithmic Sobolev inequalities and entropic convergence

Patrick Cattiaux (1, 2) , Ivan Gentil (3) , Arnaud Guillin (4)

1 MODAL'X - Modélisation aléatoire de Paris X
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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.

Domaines: Probabilités [math.PR]

Patrick Cattiaux, Ivan Gentil, Arnaud Guillin. Weak logarithmic Sobolev inequalities and entropic convergence. Probability Theory and Related Fields, 2007, 139 (3-4), pp.563-603. ⟨hal-00013700v2⟩