Some drawbacks of finite modified logarithmic Sobolev inequalilities

Laurent Miclo (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Classically, finite modified logarithmic Sobolev inequalities are used to deduce a differential inequality for the evolution of the relative entropy with respect to the invariant measure. We will check that these inequalities are ill-behaved with respect, on one hand, to the symmetrization procedure, and on the other hand, to an intersection method. The latter, introduced here for Poincaré's inequalities, enables to estimate the spectral gap of a finite reversible Markov generator L in terms of the spectral gap of the restrictions of L on two covering sets whose intersection is not empty.

Mots clés: finite irreducible Markov generator; modified logarithmic Sobolev inequality; symmetrization; Poincaré's inequality; intersection method

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Laurent Miclo. Some drawbacks of finite modified logarithmic Sobolev inequalilities. 2017. ⟨hal-01435470v1⟩