Some drawbacks of finite modified logarithmic Sobolev inequalities

Abstract

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.

Reference

Laurent Miclo, “Some drawbacks of finite modified logarithmic Sobolev inequalities”, Mathematica Scandinavica, vol. 123, n. 1, 2018, pp. 147–159.