Representation formula for the entropy and functional inequalities
Lehec, Joseph
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013), p. 885-899 / Harvested from Numdam
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On démontre une formule stochastique pour l'entropie relative par rapport à la Gaussienne, dans le genre de la formule de Borell pour la transformée de Laplace. Cette formule donne des preuves simples d'un certain nombre d'inégalités fonctionnelles.

We prove a stochastic formula for the Gaussian relative entropy in the spirit of Borell's formula for the Laplace transform. As an application, we give simple proofs of a number of functional inequalities.

Publié le : 2013-01-01
DOI : https://doi.org/10.1214/11-AIHP464
Classification:  39B62,  60J65 
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     author = {Lehec, Joseph},
     title = {Representation formula for the entropy and functional inequalities},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {49},
     year = {2013},
     pages = {885-899},
     doi = {10.1214/11-AIHP464},
     mrnumber = {3112438},
     zbl = {1279.39011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2013__49_3_885_0}
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Lehec, Joseph. Representation formula for the entropy and functional inequalities. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) pp. 885-899. doi : 10.1214/11-AIHP464. http://gdmltest.u-ga.fr/item/AIHPB_2013__49_3_885_0/

[1] K. Ball. Convex geometry and functional analysis. In Handbook of the Geometry of Banach Spaces, Vol. 1 161-194. W. B. Johnson and J. Lindenstrauss (Eds). North-Holland, Amsterdam, 2001. | MR 1863692 | Zbl 1017.46004

[2] F. Barthe. On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134 (1998) 335-361. | MR 1650312 | Zbl 0901.26010

[3] F. Barthe and N. Huet. On Gaussian Brunn-Minkowski inequalities. Studia Math. 191 (2009) 283-304. | MR 2481898 | Zbl 1166.60014

[4] F. Baudoin. Conditioned stochastic differential equations: Theory, examples and application to finance. Stochastic Process. Appl. 100 (2002) 109-145. | MR 1919610 | Zbl 1058.60040

[5] C. Borell. Diffusion equations and geometric inequalities. Potential Anal. 12 (2000) 49-71. | MR 1745333 | Zbl 0976.60065

[6] M. Boué and P. Dupuis. A variational representation for certain functionals of Brownian motion. Ann. Probab. 26 (1998) 1641-1659. | MR 1675051 | Zbl 0936.60059

[7] H. J. Brascamp and E. H. Lieb. Best constants in Young's inequality, its converse and its generalization to more than three functions. Adv. Math. 20 (1976) 151-173. | MR 412366 | Zbl 0339.26020

[8] M. Capitaine, E. P. Hsu and M. Ledoux. Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron. Commun. Probab. 2 (1997) 71-81. | MR 1484557 | Zbl 0890.60045

[9] E. Carlen and D. Cordero-Erausquin. Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities. Geom. Funct. Anal. 19 (2009) 373-405. | MR 2545242 | Zbl 1231.26015

[10] D. Cordero-Erausquin and M. Ledoux. The geometry of Euclidean convolution inequalities and entropy. Proc. Amer. Math. Soc. 138 (2010) 2755-2769. | MR 2644890 | Zbl 1196.42007

[11] A. Dembo, T. M. Cover and J. A. Thomas. Information theoretic inequalities. IEEE Trans. Inform. Theory 37 (1991) 1501-1518. | MR 1134291 | Zbl 0741.94001

[12] D. Feyel and A. S. Üstünel. Measure transport on Wiener space and the Girsanov theorem. C. R. Math. Acad. Sci. Paris 334 (2002) 1025-1028. | MR 1913729 | Zbl 1036.60004

[13] W. H. Fleming and H. M. Soner. Controlled Markov Processes and Viscosity Solutions, 2nd edition. Stochastic Modelling and Applied Probability 25. Springer, New York, 2006. | MR 2179357 | Zbl 1105.60005

[14] H. Föllmer. An entropy approach to the time reversal of diffusion processes. In Stochastic Differential Systems (Marseille-Luminy, 1984) 156-163. Lecture Notes in Control and Inform. Sci. 69. Springer, Berlin, 1985. | MR 798318 | Zbl 0562.60083

[15] H. Föllmer. Time reversal on Wiener space. In Stochastic Processes - Mathematics and Physics (Bielefeld, 1984) 119-129. Lecture Notes in Math. 1158. Springer, Berlin, 1986. | MR 838561 | Zbl 0582.60078

[16] H. Föllmer. Random fields and diffusion processes. In École d'Été de Probabilités de Saint-Flour XV-XVII, 1985-87 101-203. Lecture Notes in Math. 1362. Springer, Berlin, 1988. | MR 983373 | Zbl 0661.60063

[17] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061-1083. | MR 420249 | Zbl 0318.46049

[18] R. S. Liptser and A. N. Shiryayev. Statistics of Random Processes, Vol. 1: General Theory. Applications of Mathematics 5. Springer, New York, 1977. | MR 474486 | Zbl 0364.60004

[19] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Probability and Its Applications. Springer, Berlin, 2006. | MR 2200233 | Zbl 0837.60050

[20] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Vol. 2: Itô Calculus. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge, 2000. | MR 1780932 | Zbl 0977.60005

[21] K. T. Sturm. On the geometry of metric measure spaces. I. Acta Math. 196 (2006) 65-131. | MR 2237206 | Zbl 1105.53035

[22] M. Talagrand. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 (1996) 587-600. | MR 1392331 | Zbl 0859.46030

[23] S. R. S. Varadhan. Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics 46. SIAM, Philadelphia, 1984. | MR 758258 | Zbl 0549.60023

[24] C. Villani. Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften 338. Springer, Berlin, 2009. | MR 2459454 | Zbl 1156.53003