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Moment measures and stability for Gaussian inequalities. (English) Zbl 1413.28025
Summary: 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 $${\mathcal F}_{\gamma}(\mu) = Ent(\mu) -\frac{1}{2} W^2_2(\mu,\nu)$$, where $$\mu\in{\mathcal P}_{\gamma}$$, $$\nu\in{\mathcal P}_{\gamma}$$, $$Ent(\mu) = \int\log(\frac{\mu}{\gamma})d\mu$$ 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 analogue 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.

##### MSC:
 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 58E99 Variational problems in infinite-dimensional spaces 60H07 Stochastic calculus of variations and the Malliavin calculus
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