2017/12/30 by Alexander V. Kolesnikov, Kolesnikov, Alexander V., Egor D. Kosov +1 · 2 citations
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1801.00140
openalex publication_date 2017/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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) = \rm Ent(μ) - (1)/(2) W22(μ, ν), where μ∈ Pγ, ν∈ Pγ, \rm Ent(μ) = ∫ log( \fracμγ) d μ 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) Kähler-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.