Elementary proof of logarithmic Sobolev inequalities for Gaussian convolutions on
[Une preuve élémentaire des inégalités de Sobolev logarithmiques pour des convolutions gaussiennes sur ]
Annales mathématiques Blaise Pascal, Tome 23 (2016) no. 1, pp. 129-140.

Dans un article de 2013, l’auteur a montré que la convolution d’une mesure à support compact sur la droite réelle avec une mesure gaussienne satisfait une inégalité de Sobolev logarithmique. Dans un article de 2014, l’auteur a donné des bornes pour les constantes optimales dans ces inégalités de Sobolev logarithmiques. Dans cet article, nous donnons une preuve élémentaire simple de ce résultat.

In a 2013 paper, the author showed that the convolution of a compactly supported measure on the real line with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). In a 2014 paper, the author gave bounds for the optimal constants in these LSIs. In this paper, we give a simpler, elementary proof of this result.

Publié le :
DOI : 10.5802/ambp.357
Classification : 26D10
Keywords: Logarithmic Sobolev inequality, convolutions
Mot clés : inégalités de Sobolev logarithmiques, circonvolutions
Zimmermann, David 1

1 Department of Mathematics University of California, San Diego 9500 Gilman Drive La Jolla, CA 92093, USA
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Zimmermann, David. Elementary proof of logarithmic Sobolev inequalities for Gaussian convolutions on $\mathbb{R}$. Annales mathématiques Blaise Pascal, Tome 23 (2016) no. 1, pp. 129-140. doi : 10.5802/ambp.357. http://archive.numdam.org/articles/10.5802/ambp.357/

[1] Bakry, D. L’hypercontractivité et son utilisation en thorie des semigroupes, Lectures on probability theory (Saint-Flour, 1992), Lecture Notes in Math., Volume 1581, Springer, Berlin, 1994, pp. 1-114

[2] Bakry, D. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, New trends in stochastic analysis, World Sci. Publ., River Edge, NJ, 1997, pp. 43-75

[3] Bakry, D.; Ledoux, M. Lévy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator, Inventiones mathematicae, Volume 123 (1996), pp. 259-281

[4] Bobkov, S.; Götze, F. Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities, J. Funct. Anal., Volume 163 (1999), pp. 1-28 | DOI

[5] Bobkov, S.; Houdré, C. Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc., Volume 129 (1995) no. 616, pp. 1-28

[6] Bobkov, S.; Tetali, P. Modified logarithmic Sobolev inequalities in discrete settings, J. Theoret. Probab., Volume 19 (2006) no. 2, pp. 289-336 | DOI

[7] Cattiaux, P.; Guillin, A.; Wu, L. A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality, Probab. Theory Relat. Fields, Volume 148 (2010), pp. 285-304 | DOI

[8] Davies, E. B. Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., Volume 109 (1987) no. 2, pp. 319-333 | DOI

[9] Davies, E. B. Heat kernels and spectral theory, Cambridge University Press, 1990, ix+197 pages

[10] Davies, E. B.; Simon, B. Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal., Volume 59 (1984), pp. 335-395 | DOI

[11] Diaconis, P.; Saloff-Coste, L. Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab., Volume 6 (1996), pp. 695-750 | DOI

[12] Gross, L.; Rothaus, O. Herbst inequalities for supercontractive semigroups, J. Math. Kyoto Univ., Volume 38 (1998) no. 2, pp. 295-318

[13] Guionnet, A.; Zegarlinski, B. Lectures on logarithmic Sobolev inequalities, Séminaire de Probabilités, XXXVI, Lecture Notes in Math., Volume 1801, Springer, Berlin, 2003, pp. 1-134

[14] Ledoux, M. Isoperimetry and Gaussian analysis, Lectures on probability theory and statistics, Lecture Notes in Math., Volume 1648, Springer, Berlin, 1996, pp. 165-294

[15] Ledoux, M. The concentration of measure phenomenon, American Mathematical Society, Providence, RI, 2001, x+181 pages

[16] Ledoux, M. A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices, Séminaire de Probabilités XXXVII, Lecture Notes in Math., Volume 1832, Springer, Berlin, 2003, pp. 360-369

[17] Villani, C. Topics in optimal transportation, American Mathematical Society, Providence, RI, 2003, xvi+370 pages

[18] Wang, F.-Y.; Wang, J. Functional inequalities for convolution probability measures (http://arxiv.org/abs/1308.1713)

[19] Yau, H.T. Logarithmic Sobolev inequality for the lattice gases with mixing conditions, Commun. Math. Phys., Volume 181 (1996), pp. 367-408 | DOI

[20] Yau, H.T. Log-Sobolev inequality for generalized simple exclusion processes, Probab. Theory Related Fields, Volume 109 (1997), pp. 507-538 | DOI

[21] Zegarlinski, B. Dobrushin uniqueness theorem and logarithmic Sobolev inequalities, J. Funct. Anal., Volume 105 (1992), pp. 77-111 | DOI

[22] Zimmermann, D. Bounds for logarithmic Sobolev constants for Gaussian convolutions of compactly supported measures (http://arxiv.org/abs/1405.2581)

[23] Zimmermann, D. Logarithmic Sobolev inequalities for mollified complactly supported measures, J. Funct. Anal., Volume 265 (2013), pp. 1064-1083 | DOI

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