an:06987424
Zbl 1413.28025
Kolesnikov, Alexander V.; Kosov, Egor D.
Moment measures and stability for Gaussian inequalities
EN
Theory Stoch. Process. 22, No. 2, 47-61 (2017).
00423578
2017
j
28C20 58E99 60H07
Gaussian inequalities; optimal transportation; K??hler-Einstein equation; moment measure
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) K??hler-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.