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Some applications of mass transport to Gaussian-type inequalities

Abstract : As discovered by Brenier, mapping through a convex gradient gives the optimal transport in R-n. In the present article, this map is used in the setting of Gaussian-like measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic Sobolev and transport inequalities are recovered. Finally, a result of Caffarelli on the Brenier map is used to obtain Gaussian correlation inequalities.
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Dario Cordero-Erausquin. Some applications of mass transport to Gaussian-type inequalities. Archive for Rational Mechanics and Analysis, Springer Verlag, 2002, 161 (3), pp.257--269. ⟨10.1007/s002050100185⟩. ⟨hal-00693655⟩



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