Some drawbacks of finite modified logarithmic Sobolev inequalities

Résumé

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 the umbrella sampling procedure for Poincaré inequalities. A short spectral proof of the latter method is given 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 subsets whose union is the whole state space and whose intersection is not empty.

Référence

Laurent Miclo, « Some drawbacks of finite modified logarithmic Sobolev inequalities », Mathematica Scandinavica, vol. 123, n° 1, 2018, p. 147–159.