Some drawbacks of finite modified logarithmic Sobolev inequalities
DOI:
https://doi.org/10.7146/math.scand.a-105279Abstract
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.
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