Comptes Rendus
Probability Theory
Clark formula and logarithmic Sobolev inequalities for Bernoulli measures
[Formule de Clark et inégalités de Sobolev logarithmiques pour les mesures de Bernoulli]
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/

[1] C. Ané; M. Ledoux On logarithmic Sobolev inequalities for continuous time random walks on graphs, Probab. Theory Related Fields, Volume 116 (2000) no. 4, pp. 573-602

[2] S.G. Bobkov; M. Ledoux On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures, J. Funct. Anal., Volume 156 (1998) no. 2, pp. 347-365

[3] M. Capitaine; E.P. Hsu; M. Ledoux Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces, Electron. Comm. Probab., Volume 2 (1997), pp. 71-81 (electronic)

[4] P. Dai Pra; A.M. Paganoni; G. Posta Entropy inequalities for unbounded spin systems, Ann. Probab., Volume 30 (2002) no. 4, pp. 1959-1976

[5] F. Gao, J. Quastel. Exponential decay of entropy in the random transposition and Bernoulli–Laplace models, Preprint, 2002

[6] H. Holden; T. Lindstrøm; B. Øksendal; J. Ubøe Discrete Wick calculus and stochastic functional equations, Potential Anal., Volume 1 (1992) no. 3, pp. 291-306

[7] M. Leitz-Martini, A discrete Clark–Ocone formula, Maphysto Research Report No 29, 2000

[8] P.A. Meyer Quantum Probability for Probabilists, Lecture Notes in Math., 1538, Springer-Verlag, 1993

[9] N. Privault; W. Schoutens Discrete chaotic calculus and covariance identities, Stochastics Stochastics Rep., Volume 72 (2002), pp. 289-315 (Eurandom Report 006, 2000)

[10] L. Wu A new modified logarithmic Sobolev inequality for Poisson point processes and several applications, Probab. Theory Related Fields, Volume 118 (2000) no. 3, pp. 427-438

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This research was supported by the National Natural Science Foundation of China under grant No. 19971025.

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