Moment measures and stability for Gaussian inequalities

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 ${\mathcal{F}}_{\gamma }(\mu )=\mathrm{E}\mathrm{n}\mathrm{t}(\mu )-\frac{1}{2}{W}_2^2(\mu ,\nu )$, where $\mu \in {\mathcal{P}}_{\gamma },\nu \in {\mathcal{P}}_{\gamma }$, ${\rm Ent}(\mu) = \int \log\bigl( \frac{\mu}{\gamma}\bigr) d \mu$ 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 analog of the (real) K"ahler-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.