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Theory Stoch. Process., 2017, Volume 22(38), Issue 2, Pages 47–61 (Mi thsp179)  

Moment measures and stability for Gaussian inequalities

Alexander V. Kolesnikova, Egor D. Kosovb

a National Research University "Higher School of Economics" Moscow, Russia
b Departament of Mechanics and Mathematics, Moscow State University, 119991 Moscow, Russia; National Research University Higher School of Economics, Moscow, Russia

Abstract: Let $\gamma$ be the standard Gaussian measure on $\mathbb{R}^n$ and let $\mathcal{P}_{\gamma}$ be the space of probability measures that are absolutely continuous with respect to $\gamma$. We study lower bounds for the functional $\mathcal{F}_{\gamma}(\mu) = Ent(\mu) - \frac{1}{2} W^2_2(\mu, \nu)$, where $\mu \in \mathcal{P}_{\gamma}, \nu \in \mathcal{P}_{\gamma}$, $Ent(\mu) = \int \log( \frac{\mu}{\gamma}) d \mu$ is the relative Gaussian entropy, and $W_2$ is the quadratic Kantorovich distance. The minimizers of $\mathcal{F}_{\gamma}$ are solutions to a dimension-free Gaussian analog of the (real) Kähler–Einstein equation. We show that $\mathcal{F}_{\gamma}(\mu) $ is bounded from below under the assumption that the Gaussian Fisher information of $\nu$ is finite and prove a priori estimates for the minimizers. Our approach relies on certain stability estimates for the Gaussian log-Sobolev and Talagrand transportation inequalities.

Keywords: Gaussian inequalities, optimal transportation, Kähler-Einstein equation, moment measure.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-11-01058
This research has been supported by the Russian Science Foundation Grant N 17-11-01058 (at Moscow Lomonosov State University)


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Bibliographic databases:
MSC: 28C20, 58E99, 60H07
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Citation: Alexander V. Kolesnikov, Egor D. Kosov, “Moment measures and stability for Gaussian inequalities”, Theory Stoch. Process., 22(38):2 (2017), 47–61

Citation in format AMSBIB
\Bibitem{KolKos17}
\by Alexander~V.~Kolesnikov, Egor~D.~Kosov
\paper Moment measures and stability for Gaussian inequalities
\jour Theory Stoch. Process.
\yr 2017
\vol 22(38)
\issue 2
\pages 47--61
\mathnet{http://mi.mathnet.ru/thsp179}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3843524}
\zmath{https://zbmath.org/?q=an:06987424}


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