Nonlinear transformations of convex measures

V. I. Bogachev   ; A. V. Kolesnikov

Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 1, pp. 27-51

Résumé

Given a uniformly convex measure $\mu$ on $R^\infty$ that is equivalent to its translation to the vector $(1,0,0,\ldots)$ and a probability measure $\nu$ that is absolutely continuous with respect to $\mu$, we show that there is a Borel mapping $T=(T_k)_{k=1}^\infty$ of $R^\infty$ transforming $\mu$ into $\nu$ and having the form $T(x)=x+F(x)$, where $F$ has values in $l^2$. Moreover, if $\mu$ is a product-measure, then $T$ can be chosen triangular in the sense that each component $T_k$ is a function of $x_1,\dots,x_k$. In addition, for any uniformly convex measure $\mu$ on $R^\infty$ and any probability measure $\nu$ with finite entropy $\textrm{ent}_\mu(\nu)$ with respect to $\mu$, the canonical triangular mapping $T=I+F$ transforming $\mu$ into $\nu$ satisfies the inequality $\|F\|_{L^2(\mu,l^2)}^2\le C(\mu)\textrm{ent}_\mu (\nu)$. Several inverse assertions are proved. Our results apply, in particular, to the standard Gaussian product-measure. As an application we obtain a new sufficient condition for the absolute continuity of a nonlinear image of a convex measure and the membership of the corresponding Radon–Nikodym derivative in the class $L\log L$.

Détail

Keywords: convex measure, Gaussian measure, product-measure, absolute continuity, triangular mapping.
Mots-clés : Cameron–Martin space

Bibliographie

[1] Bogachev V. I., “Differentiable measures and the Malliavin calculus”, J. Math. Sci. (New York), 87:4 (1997), 3577–3731 | DOI | MR | Zbl

[2] Bogachev V. I., Gaussovskie mery, Nauka, M., 1997, 352 pp. | MR

[3] Bogachev V. I., Osnovy teorii mery, v. 1, 2, Regulyarnaya i khaoticheskaya dinamika, Moskva, Izhevsk, 2003

[4] Üstünel A. S., Zakai M., Transformation of Measure on Wiener Space, Springer-Verlag, Berlin, 2000, 296 pp. | MR

[5] Shepp L. A., “Distinguishing a sequence of random variables from a translate of itself”, Ann. Math. Statist., 36 (1965), 1107–1112 | DOI | MR | Zbl

[6] Bogachev V. I., Kolesnikov A. V., “Nelineinye preobrazovaniya vypuklykh mer i entropiya plotnostei Radona–Nikodima”, Dokl. RAN, 397:2 (2004), 155–159 | MR

[7] Kolesnikov A. V., “Neravenstva vypuklosti i nelineinye preobrazovaniya mer”, Dokl. RAN, 396:3 (2004), 300–304 | MR | Zbl

[8] Bogachev V. I., Kolesnikov A. V., Medvedev K. V., “O treugolnykh preobrazovaniyakh mer”, Dokl. RAN, 396:6 (2004), 727–732 | MR | Zbl

[9] Bogachev V. I., Kolesnikov A. V., Medvedev K. V., “Treugolnye preobrazovaniya mer”, Matem. sb., 196:3 (2005), 3–30 | MR | Zbl

[10] Kolesnikov A. V., “Convexity inequalities and optimal transport of infinite-dimensional measures”, J. Math. Pures Appl., 83:11 (2004), 1373–1404 | MR | Zbl

[11] Knothe H., “Contributions to the theory of convex bodies”, Michigan Math. J., 4 (1957), 39–52 | DOI | MR | Zbl

[12] Talagrand M., “Transportation cost for Gaussian and other product measures”, Geom. Funct. Anal., 6:3 (1996), 587–600 | DOI | MR | Zbl

[13] Bobkov S. G., “Large deviations via transference plans”, Adv. Math. Res., 2 (2003), 151–175 | MR | Zbl

[14] Otto F., Villani C., “Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality”, J. Funct. Anal., 173:2 (2000), 361–400 | DOI | MR | Zbl

[15] Cordero-Erausquin D., “Some applications of mass transport to Gaussian-type inequalities”, Arch. Rational Mech. Anal., 161:3 (2002), 257–269 | DOI | MR | Zbl

[16] Rachev S. T., Rüschendorf L., Mass Transportation Problems, v. I, Springer-Verlag, New York, 1998, 508 pp. | MR

[17] Fernique X., “Extension du théorème de Cameron-Martin aux translations aléatoires”, Ann. Probab., 31:3 (2003), 1296–1304 | DOI | MR | Zbl

[18] Feyel D., Üstünel A. S., “Measure transport on Wiener space and the Girsanov theorem”, C. R. Acad. Sci. Paris, 334:11 (2002), 1025–1028 | MR | Zbl

[19] Feyel D., Üstünel A. S., “Monge–Kantorovitch measure transportation and Monge-Ampère equation on Wiener space”, Probab. Theory Related Fields, 128:3 (2004), 347–385 | DOI | MR | Zbl

[20] Ledoux M., The Concentration of Measure Phenomenon, Amer. Math. Soc., Providence, RI, 2001, 181 pp. | MR | Zbl

[21] Aleksandrova D. E., “Skhodimost treugolnykh preobrazovanii mer”, Teoriya veroyatn. i ee primen., 50:1 (2005), 145–150 | MR

[22] Borell C., “A note on conditional probabilities of a convex measure”, Lecture Notes in Phys., 77, 1978, 68–72 | MR

[23] Üstünel A. S., Zakai M., “Measures induced on Wiener space by monotone shifts”, Probab. Theory Related Fields, 105:4 (1996), 545–563 | DOI | MR | Zbl

[24] Brenier Y., “Polar factorization and monotone rearrangement of vector-valued functions”, Comm. Pure Appl. Math., 44:4 (1991), 375–417 | DOI | MR | Zbl

[25] McCann R. J., “Existence and uniqueness of monotone measure-preserving maps”, Duke Math. J., 80:2 (1995), 309–323 | DOI | MR | Zbl

[26] Üstünel A. S., Zakai M., “On the uniform integrability of the Radon–Nikodým densities for Wiener measure”, J. Funct. Anal., 159:2 (1998), 642–663 | DOI | MR | Zbl

[27] Albeverio S., Kondratiev Yu. G., Röckner M., “Dirichlet operators via stochastic analysis”, J. Funct. Anal., 128:1 (1995), 102–138 | DOI | MR | Zbl

[28] Kulik A. M., “Log-Sobolev inequality, exponential integrability and large deviation estimates for $C(\alpha,\beta)$ log-concave measures”, Random Oper. Stochastic Equations, 10:2 (2002), 105–122 | DOI | MR | Zbl

[29] Kulik A. M., Pilipenko A. Yu., “Nelineinye preobrazovaniya gladkikh mer na beskonechnomernykh prostranstvakh”, Ukr. matem. zhurn., 52:9 (2000), 1226–1250 | MR | Zbl

[30] Bobkov S. G., Gentil J., Ledoux M., “Hypercontractivity of Hamilton–Jacobi equations”, J. Math. Pures Appl., 80:7 (2001), 669–696 | DOI | MR | Zbl