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.
@article{AMBP_2016__23_1_129_0,
author = {David Zimmermann},
title = {Elementary proof of logarithmic {Sobolev} inequalities for {Gaussian} convolutions on $\mathbb{R}$},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {129--140},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {23},
number = {1},
year = {2016},
doi = {10.5802/ambp.357},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.357/}
}
TY - JOUR
AU - David Zimmermann
TI - Elementary proof of logarithmic Sobolev inequalities for Gaussian convolutions on $\mathbb{R}$
JO - Annales mathématiques Blaise Pascal
PY - 2016
SP - 129
EP - 140
VL - 23
IS - 1
PB - Annales mathématiques Blaise Pascal
UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.357/
DO - 10.5802/ambp.357
LA - en
ID - AMBP_2016__23_1_129_0
ER -
%0 Journal Article
%A David Zimmermann
%T Elementary proof of logarithmic Sobolev inequalities for Gaussian convolutions on $\mathbb{R}$
%J Annales mathématiques Blaise Pascal
%D 2016
%P 129-140
%V 23
%N 1
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.357/
%R 10.5802/ambp.357
%G en
%F AMBP_2016__23_1_129_0
David Zimmermann. 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. https://ambp.centre-mersenne.org/articles/10.5802/ambp.357/
[1] 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] 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] Lévy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator, Inventiones mathematicae, Volume 123 (1996), pp. 259-281
[4] Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities, J. Funct. Anal., Volume 163 (1999), pp. 1-28 | DOI
[5] Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc., Volume 129 (1995) no. 616, pp. 1-28
[6] Modified logarithmic Sobolev inequalities in discrete settings, J. Theoret. Probab., Volume 19 (2006) no. 2, pp. 289-336 | DOI
[7] A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality, Probab. Theory Relat. Fields, Volume 148 (2010), pp. 285-304 | DOI
[8] Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., Volume 109 (1987) no. 2, pp. 319-333 | DOI
[9] Heat kernels and spectral theory, Cambridge University Press, 1990, ix+197 pages
[10] Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal., Volume 59 (1984), pp. 335-395 | DOI
[11] Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab., Volume 6 (1996), pp. 695-750 | DOI
[12] Herbst inequalities for supercontractive semigroups, J. Math. Kyoto Univ., Volume 38 (1998) no. 2, pp. 295-318
[13] Lectures on logarithmic Sobolev inequalities, Séminaire de Probabilités, XXXVI, Lecture Notes in Math., Volume 1801, Springer, Berlin, 2003, pp. 1-134
[14] Isoperimetry and Gaussian analysis, Lectures on probability theory and statistics, Lecture Notes in Math., Volume 1648, Springer, Berlin, 1996, pp. 165-294
[15] The concentration of measure phenomenon, American Mathematical Society, Providence, RI, 2001, x+181 pages
[16] 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] Topics in optimal transportation, American Mathematical Society, Providence, RI, 2003, xvi+370 pages
[18] Functional inequalities for convolution probability measures (http://arxiv.org/abs/1308.1713)
[19] Logarithmic Sobolev inequality for the lattice gases with mixing conditions, Commun. Math. Phys., Volume 181 (1996), pp. 367-408 | DOI
[20] Log-Sobolev inequality for generalized simple exclusion processes, Probab. Theory Related Fields, Volume 109 (1997), pp. 507-538 | DOI
[21] Dobrushin uniqueness theorem and logarithmic Sobolev inequalities, J. Funct. Anal., Volume 105 (1992), pp. 77-111 | DOI
[22] Bounds for logarithmic Sobolev constants for Gaussian convolutions of compactly supported measures (http://arxiv.org/abs/1405.2581)
[23] Logarithmic Sobolev inequalities for mollified complactly supported measures, J. Funct. Anal., Volume 265 (2013), pp. 1064-1083 | DOI
Cité par Sources :