Moment measures and stability for Gaussian inequalities
Let γ be the standard Gaussian measure on Rn and let Pγ be the space of probability measures that are absolutely continuous with respect to γ. We study lower bounds for the functional Fγ(µ) = Ent(µ) − 1 2W2 2 (µ, ν), where µ ∈ Pγ, ν ∈ Pγ, Ent(µ) = R log µ γ dµ is the relative Gaussian entropy, and W2 is the quadratic Kantorovich distance. The minimizers of Fγ are solutions to a dimension-free Gaussian analog of the (real) K¨ahler–Einstein equation. We show that Fγ(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.