Some drawbacks of finite modified logarithmic Sobolev inequalities

  • Laurent Miclo

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.

References

Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., and Scheffer, G., Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses, vol. 10, Société Mathématique de France, Paris, 2000.

Madras, N. and Randall, D., Markov chain decomposition for convergence rate analysis, Ann. Appl. Probab. 12 (2002), no. 2, 581–606. https://doi.org/10.1214/aoap/1026915617

>

Miclo, L., Remarques sur l'hypercontractivité et l'évolution de l'entropie pour des chaînes de Markov finies, in “Séminaire de Probabilités, XXXI”, Lecture Notes in Math., vol. 1655, Springer, Berlin, 1997, pp. 136--167. https://doi.org/10.1007/BFb0119300

>

Saloff-Coste, L., Lectures on finite Markov chains, in “Lectures on probability theory and statistics (Saint-Flour, 1996)'', Lecture Notes in Math., vol. 1665, Springer, Berlin, 1997, pp. 301--413. https://doi.org/10.1007/BFb0092621

>

Wu, L., A new modified logarithmic Sobolev inequality for Poisson point processes and several applications, Probab. Theory Related Fields 118 (2000), no. 3, 427–438. https://doi.org/10.1007/PL00008749

>

Published
2018-08-06
How to Cite
Miclo, L. (2018). Some drawbacks of finite modified logarithmic Sobolev inequalities. MATHEMATICA SCANDINAVICA, 123(1), 147-159. https://doi.org/10.7146/math.scand.a-105279
Section
Articles