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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 3, Pages 433–456 (Mi tvp87)

Integrability of absolutely continuous measure transformations and applications to optimal transportation

V. I. Bogachev, A. V. Kolesnikov

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

Abstract: Given two Borel probability measures $\mu$ and $\nu$ on $\mathbf{R}^d$ such that $d\nu/d\mu =g$, we consider certain mappings of the form $T(x)=x+F(x)$ that transform $\mu$ into $\nu$. Our main results give estimates of the form $\int_{\mathbf{R}^d}\Phi_1(|F|) d\mu\leq\int_{\mathbf{R}^d}\Phi_2(g) d\mu$ for certain functions $\Phi_1$ and $\Phi_2$ under appropriate assumptions on $\mu$. Applications are given to optimal mass transportations in the Monge problem.

Keywords: optimal transportation, Gaussian measure, convex measure, logarithmic Sobolev inequality, Poincaré, inequality, Talagrand inequality.

DOI: https://doi.org/10.4213/tvp87

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English version:
Theory of Probability and its Applications, 2006, 50:3, 367–385

Bibliographic databases:

Citation: V. I. Bogachev, A. V. Kolesnikov, “Integrability of absolutely continuous measure transformations and applications to optimal transportation”, Teor. Veroyatnost. i Primenen., 50:3 (2005), 433–456; Theory Probab. Appl., 50:3 (2006), 367–385

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp87
• https://doi.org/10.4213/tvp87
• http://mi.mathnet.ru/eng/tvp/v50/i3/p433

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This publication is cited in the following articles:
1. Shaposhnikov S.V., “Positiveness of invariant measures of diffusion processes”, Dokl. Math., 76:1 (2007), 533–538
2. A. V. Kolesnikov, “Sobolev regularity of transportation of probability measures and transportation inequalities”, Theory Probab. Appl., 57:2 (2013), 243–264
3. V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890
4. Bogachev V.I., Kolesnikov A.V., “Sobolev Regularity for the Monge-Ampere Equation in the Wiener Space”, Kyoto J. Math., 53:4 (2013), 713–738
5. Alexander V. Kolesnikov, Egor D. Kosov, “Moment measures and stability for Gaussian inequalities”, Theory Stoch. Process., 22(38):2 (2017), 47–61
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