[Formule de Clark et inégalités de Sobolev logarithmiques pour les mesures de Bernoulli]
A l'aide d'une formule de Clark pour la représentation prévisible de variables aléatoire en temps discret et en adaptant la preuve présentée dans [Electron. Commun. Probab. 2 (1997) 71–81] dans le cas brownien, nous obtenons une preuve des inégalités de Sobolev logarithmiques (inégalité modifiée et inégalité L1) pour les mesures de Bernoulli. Nous présentons aussi une borne qui améliore ces inégalités ainsi que l'inégalité de constante optimale de [J. Funct. Anal. 156 (2) (1998) 347–365].
Using a Clark formula for the predictable representation of random variables in discrete time and adapting the method presented in [Electron. Commun. Probab. 2 (1997) 71–81] in the Brownian case, we obtain a proof of modified and L1 logarithmic Sobolev inequalities for Bernoulli measures. We also prove a bound that improves these inequalities as well as the optimal constant inequality of [J. Funct. Anal. 156 (2) (1998) 347–365].
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@article{CRMATH_2003__336_1_51_0,
author = {Gao, Fuqing and Privault, Nicolas},
title = {Clark formula and logarithmic {Sobolev} inequalities for {Bernoulli} measures},
journal = {Comptes Rendus. Math\'ematique},
pages = {51--56},
publisher = {Elsevier},
volume = {336},
number = {1},
year = {2003},
doi = {10.1016/S1631-073X(02)00014-6},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)00014-6/}
}
TY - JOUR AU - Gao, Fuqing AU - Privault, Nicolas TI - Clark formula and logarithmic Sobolev inequalities for Bernoulli measures JO - Comptes Rendus. Mathématique PY - 2003 SP - 51 EP - 56 VL - 336 IS - 1 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)00014-6/ DO - 10.1016/S1631-073X(02)00014-6 LA - en ID - CRMATH_2003__336_1_51_0 ER -
%0 Journal Article %A Gao, Fuqing %A Privault, Nicolas %T Clark formula and logarithmic Sobolev inequalities for Bernoulli measures %J Comptes Rendus. Mathématique %D 2003 %P 51-56 %V 336 %N 1 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)00014-6/ %R 10.1016/S1631-073X(02)00014-6 %G en %F CRMATH_2003__336_1_51_0
Gao, Fuqing; Privault, Nicolas. Clark formula and logarithmic Sobolev inequalities for Bernoulli measures. Comptes Rendus. Mathématique, Tome 336 (2003) no. 1, pp. 51-56. doi : 10.1016/S1631-073X(02)00014-6. http://archive.numdam.org/articles/10.1016/S1631-073X(02)00014-6/
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☆ This research was supported by the National Natural Science Foundation of China under grant No. 19971025.
