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 Theory Stoch. Process.: Year: Volume: Issue: Page: Find

 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}