Comptes Rendus
no. 1
Probability Theory
Clark formula and logarithmic Sobolev inequalities for Bernoulli measures
[Formule de Clark et inégalités de Sobolev logarithmiques pour les mesures de Bernoulli]

Présenté par : Paul Malliavin

Fuqing Gao1 ; Nicolas Privault2
1 Department of Mathematics, Wuhan University, 430072 Wuhan, PR China
2 Département de mathématiques, Université de La Rochelle, 17042 La Rochelle, France
Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 51-56.

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].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)00014-6

Fuqing Gao 1 ; Nicolas Privault 2

1 Department of Mathematics, Wuhan University, 430072 Wuhan, PR China
2 Département de mathématiques, Université de La Rochelle, 17042 La Rochelle, France 
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Fuqing Gao; Nicolas Privault. Clark formula and logarithmic Sobolev inequalities for Bernoulli measures. Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 51-56. doi : 10.1016/S1631-073X(02)00014-6. https://comptes-rendus.academie-sciences.fr/mathematique/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.

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