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Moment measures and stability for Gaussian inequalities. (English) Zbl 1413.28025
Summary: 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 analogue 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.

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
58E99 Variational problems in infinite-dimensional spaces
60H07 Stochastic calculus of variations and the Malliavin calculus
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