Abstract.
This paper deals with a generalization of a result due to Brascamp and Lieb which states that in the space of probabilities with log-concave density with respect to a Gaussian measure on this Gaussian measure is the one which has strongest moments. We show that this theorem remains true if we replace xα by a general convex function. Then, we deduce a correlation inequality for convex functions quite better than the one already known. Finally, we prove results concerning stochastic analysis on abstract Wiener spaces through the notion of approximate limit.
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Bakry, D., Michel, Sur les inégalités, F.K.G.: Séminaire de Probabilités XXVI. Lecture Notes in Math. 1526, 170–188 (1990)
Ben Arous, G., Gradinaru, M., Ledoux, M.: Hölder norms and the support theorem for diffusions. Annales de l’I.H.P., Proba. et Stat. 30 (3), 415–436 (1994)
Bogachev, V.I.: The Onsager-Machlup functions for Gaussian measures. Doklady Math. 52 (2), 216–218 (1995)
Borell, C.: A note on Gauss measures which agree on small balls. Ann. Inst. H. Poincaré 13, 231–238 (1977)
Brenier, Y.: Polar factorization and monotone rearrangement of vector valued functions. Comm. Pure Appl. Math. 44, 375–417 (1991)
Brascamp, H.J., Lieb, E.H.: On extensions of Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log-concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22, 366–389 (1976)
Caffarelli, L.A.: Monotinicity properties of optimal transportation and the FKG and related inequalities. Commun. Math. Phys. 214 (3), 547–563 (2000)
Caffarelli, L.A.: The regularity of mappings with a convex potential. J. A. M. S. 5, 99–104 (1992)
Caffarelli, L.A.: Boundary regularity of maps with convex potential. Comm. Pure Appl. Math. 45, 1141–1151 (1992)
Carmona, R.A., Nualart, D.: Traces of random variables on Wiener space and Onsager-Machlup Functional. J. Funct. Anal. 107 (2), 402–438 August 1, 1992
Cordero-Erausquin, D.: Some applications of mass transport to Gaussian type inequalities. Arch. Rational Mech. Anal. 161, 257–269 (2002)
Fang, S.: Sur la continuité approximative forte des fonctionnelles d’Itô. Stochastics and Stochastics Reports 36, 193–204 (1991)
Gross, L.: Measurable Functions on Hilbert space. Trans. Am. Math. Soc. 105, 372–390 (1962)
Gross, L.: Abstract Wiener spaces. Proceedings of the Fifth Berkeley Symp. Math. Statist. Prob. II, part I, 31–42 (1967)
Hargé, G.: A particular case of correlation inequality for the Gaussian measure. Ann. Probab. 27 (4), 1939–1951 (1999)
Hargé, G.: Limites approximatives sur l’espace de Wiener. Potential Anal. 16 (2), 169–191 (2002)
Hu, Y.: Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities. J. Th. Probab. 10, 835–848 (1997)
Hu, Y.Z., Meyer, P.A.: Sur les intégrales multiples de Stratonovitch. Séminaire de probabilités XXII, 72–81
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, 1981
Johnson, G.W., Kallianpur, G.: Homogeneous chaos, p-forms, scaling and the Feynman integral. Transactions of the A.M.S. 340 (2), 503–548 (1993)
Kuo, H.H.: Gaussian measures in Banach spaces. Lecture Notes in Math. 463, Springer, 1975
Mayer-Wolf, E., Zeitouni, O.: Onsager Machlup functionals for non trace class SPDE’S. Proba. Th. Rel. Fields 95, 199–216 (1993)
McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309–323 (1995)
Millet, A., Nualart, D.: Support theorems for a class of anticipating stochastic differential equations. Stochastics and Stochastics Reports 39, 1–24 (1992)
Nualart, D.: The Malliavin Calculus and Related Topics. Springer-Verlag, Berlin, Heidelberg, New-York, 1995
Nualart, D., Zakai, M.: Generalized Stochastic Integrals and the Malliavin Calculus. Proba. Th. Rel. Fields 73, 255–280 (1986)
Pitt, L.D.: A gaussian correlation inequality for symetric convex sets. Ann. Probab. 5, 470–474 (1977)
Rosinski, J.: On Stochastic Integration by Series of Wiener Integrals. Appl. Math. Optim. 19, 137–155 (1989)
Royer, G.: Une initiation aux inégalités de Sobolev logarithmiques. Société Mathématique de France, Cours spécialisé 5, (1999)
Schechtman, G., Schlumprecht, T., Zinn, J.: On the Gaussian measure of the intersection of symmetric convex sets. Ann. Probab. 26, 346–357 (1998)
Shepp, L.A., Zeitouni, O.: A note on conditional exponential moments and Onsager Machlup Functionals. Ann. Probab. 20, 652–654 (1992)
Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principal. Sixth Berkeley Symp. Math. Statist. Prob., 333–359 (1972)
Sugita, H.: Hu-Meyer’s mutiple Stratonovich integral and essential continuity of multiple Wiener integral. Bull. Sc. Math. 113, 463–474 (1989)
Sugita, H.: Various topologies in the Wiener space and Lévy’s stochastic area. Proba. Th. Rel. Fields 91 (3–4), 283–296 (1992)
Szarek, S., Werner, E.: A nonsymmetric correlation inequality for Gaussian measure. J. Multivariate Anal. 68, 193–211 (1999)
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Mathematics Subject Classification (2000): Primary: 28C20, 60E15, 60H05
Revised version: 20 February 2004
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Hargé, G. A convex/log-concave correlation inequality for Gaussian measure and an application to abstract Wiener spaces. Probab. Theory Relat. Fields 130, 415–440 (2004). https://doi.org/10.1007/s00440-004-0365-8
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DOI: https://doi.org/10.1007/s00440-004-0365-8