Geodesic
Integrability of absolutely continuous measure transformations and applications to optimal transportation
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 3, pp. 433-456

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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.
Mots-clés : optimal transportation
Keywords: Gaussian measure, convex measure, logarithmic Sobolev inequality, Poincaré, inequality, Talagrand inequality. 
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     author = {V. I. Bogachev and A. V. Kolesnikov},
     title = {Integrability of absolutely continuous measure transformations and applications to optimal transportation},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {433--456},
     publisher = {mathdoc},
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     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2005_50_3_a1/}
}
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V. I. Bogachev; A. V. Kolesnikov. Integrability of absolutely continuous measure transformations and applications to optimal transportation. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 3, pp. 433-456. http://geodesic.mathdoc.fr/item/TVP_2005_50_3_a1/