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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
Аннотация:
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 $\mathcal{F}_{\gamma}(\mu) = {\rm Ent}(\mu) - \frac{1}{2} W^2_2(\mu, \nu)$, where $\mu \in \mathcal{P}_{\gamma}, \nu \in \mathcal{P}_{\gamma}$, ${\rm Ent}(\mu) = \int \log\bigl( \frac{\mu}{\gamma}\bigr) d \mu$ 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.
Ключевые слова:
Gaussian inequalities, optimal transportation, Kähler-Einstein equation, moment measure.
Образец цитирования:
Alexander V. Kolesnikov, Egor D. Kosov, “Moment measures and stability for Gaussian inequalities”, Theory Stoch. Process., 22(38):2 (2017), 47–61
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/thsp179 https://www.mathnet.ru/rus/thsp/v22/i2/p47
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