Stein kernels and moment maps

Abstract

We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge-Ampère equation. As a consequence, we show how regularity bounds in certain weighted Sobolev spaces on these maps control the rate of convergence in the classical central limit theorem, and derive new rates in Kantorovitch-Wasserstein distance in the log-concave situation, with explicit polynomial dependence on the dimension.