Stein kernels, optimal transport and the central limit theorem

Stein kernels are a way to compare a given probability measure to some reference measure, typically the standard gaussian distribution, via integration by parts formulas. In this talk, I will present a construction of such kernels via optimal transport, and some applications. As a byproduct, we obtain new quantitative bounds on the rate of convergence in the central limit theorem for multidimensional log-concave distributions via regularity theory for Monge-Ampére PDEs.