### Moment measures and stability for Gaussian inequalities

**Alexander V. Kolesnikov, Egor D. Kosov**

*Theory of Stochastic Processes*

*Vol.22 (38), no.2, 2017, pp.47-61*

Let γ be the standard Gaussian measure on R

^{n}and let P

_{γ}be the space of probability measures that are absolutely continuous with respect to γ. We study lower bounds for the functional F

_{γ}(μ) = Ent(μ) - 1/2 W

_{2}

^{2}(μ, ν), where μ ∈ P

_{γ}, ν ∈ P

_{γ}, Ent(μ) = ∫ log ( μ/γ ) dμ is the relative Gaussian entropy, and W

_{2}is the quadratic Kantorovich distance. The minimizers of F

_{γ}are solutions to a dimension-free Gaussian analog of the (real) Kähler-Einstein equation. We show that F

_{γ}(μ) is bounded from below under the assumption that the Gaussian Fisher information of ν 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.