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 Rⁿ 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 W₂²(μ, ν), where μ ∈ P_γ, ν ∈ P_γ, Ent(μ) = ∫ log ( μ/γ ) dμ is the relative Gaussian entropy, and W₂ 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.