Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Representation formula for the entropy and functional inequalities

Joseph Lehec

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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.


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.

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Ann. Inst. H. Poincaré Probab. Statist. Volume 49, Number 3 (2013), 885-899.

First available in Project Euclid: 2 July 2013

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Primary: 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx] 60J65: Brownian motion [See also 58J65]

Gaussian measure Entropy Functional inequalities Girsanov’s formula


Lehec, Joseph. Representation formula for the entropy and functional inequalities. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 3, 885--899. doi:10.1214/11-AIHP464.

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  • [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.
  • [2] F. Barthe. On a reverse form of the Brascamp–Lieb inequality. Invent. Math. 134 (1998) 335–361.
  • [3] F. Barthe and N. Huet. On Gaussian Brunn–Minkowski inequalities. Studia Math. 191 (2009) 283–304.
  • [4] F. Baudoin. Conditioned stochastic differential equations: Theory, examples and application to finance. Stochastic Process. Appl. 100 (2002) 109–145.
  • [5] C. Borell. Diffusion equations and geometric inequalities. Potential Anal. 12 (2000) 49–71.
  • [6] M. Boué and P. Dupuis. A variational representation for certain functionals of Brownian motion. Ann. Probab. 26 (1998) 1641–1659.
  • [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.
  • [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.
  • [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.
  • [10] D. Cordero-Erausquin and M. Ledoux. The geometry of Euclidean convolution inequalities and entropy. Proc. Amer. Math. Soc. 138 (2010) 2755–2769.
  • [11] A. Dembo, T. M. Cover and J. A. Thomas. Information theoretic inequalities. IEEE Trans. Inform. Theory 37 (1991) 1501–1518.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [17] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061–1083.
  • [18] R. S. Liptser and A. N. Shiryayev. Statistics of Random Processes, Vol. 1: General Theory. Applications of Mathematics 5. Springer, New York, 1977.
  • [19] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Probability and Its Applications. Springer, Berlin, 2006.
  • [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.
  • [21] K. T. Sturm. On the geometry of metric measure spaces. I. Acta Math. 196 (2006) 65–131.
  • [22] M. Talagrand. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 (1996) 587–600.
  • [23] S. R. S. Varadhan. Large Deviations and Applications. CBMS–NSF Regional Conference Series in Applied Mathematics 46. SIAM, Philadelphia, 1984.
  • [24] C. Villani. Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften 338. Springer, Berlin, 2009.