Moment measures and stability for Gaussian inequalities

Abstract

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 ${\mathcal{F}}_{\gamma }(\mu )=\mathrm{Ent}(\mu )-\frac{1}{2}{W}_2^2(\mu ,\nu )$, where $\mu \in {\mathcal{P}}_{\gamma },\nu \in {\mathcal{P}}_{\gamma }$, $\mathrm{Ent}(\mu )=\int \log \left(\frac{\mu }{\gamma }\right)d\mu$ 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) Kähler-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.

References in corpus (2)

• Mass transportation and contractions
• Wasserstein Stability of the Entropy Power Inequality for Log-Concave Densities

Cited by in corpus (1)

• Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels