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 Mat. Sb., 2005, Volume 196, Number 3, Pages 3–30 (Mi msb1271)

Triangular transformations of measures

V. I. Bogachev, A. V. Kolesnikov, K. V. Medvedev

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A new identity for the entropy of a non-linear image of a measure on $\mathbb R^n$ is obtained, which yields the well-known Talagrand's inequality. Triangular mappings on $\mathbb R^n$ and $\mathbb R^\infty$ are studied, that is, mappings $T$ such that the $i$th coordinate function $T_i$ depends only on the variables $x_1,…,x_i$. With the help of such mappings the well-known open problem on the representability of each probability measure that is absolutely continuous with respect to a Gaussian measure $\gamma$ on an infinite dimensional space as the image of $\gamma$ under a map of the form $T(x)=x+F(x)$ where $F$ takes values in the Cameron–Martin space of the measure $\gamma$ is solved in the affirmative. As an application, a generalized logarithmic Sobolev inequality is also proved.

DOI: https://doi.org/10.4213/sm1271

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English version:
Sbornik: Mathematics, 2005, 196:3, 309–335

Bibliographic databases:

UDC: 519.2
MSC: 28C20, 46G12, 60B11

Citation: V. I. Bogachev, A. V. Kolesnikov, K. V. Medvedev, “Triangular transformations of measures”, Mat. Sb., 196:3 (2005), 3–30; Sb. Math., 196:3 (2005), 309–335

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb1271
• https://doi.org/10.4213/sm1271
• http://mi.mathnet.ru/eng/msb/v196/i3/p3

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This publication is cited in the following articles:
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3. D. E. Aleksandrova, “Convergence of triangular transformations of measures”, Theory Probab. Appl., 50:1 (2006), 113–118
4. V. I. Bogachev, A. V. Kolesnikov, “Integrability of Absolutely Continuous Transformations of Measures and Applications to Optimal Mass Transportation”, Theory Probab Appl, 50:3 (2006), 367
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