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

This article is cited in 5 scientific papers (total in 5 papers)

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

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Received: 30.05.2005

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. A. V. Kolesnikov, “Sobolev regularity of transportation of probability measures and transportation inequalities”, Theory Probab. Appl., 57:2 (2013), 243–264  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  scopus
    5. Alexander V. Kolesnikov, Egor D. Kosov, “Moment measures and stability for Gaussian inequalities”, Theory Stoch. Process., 22(38):2 (2017), 47–61  mathnet
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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