Eustasio del Barrio, Universidad de Valladolid

Abstract:

We consider the problem of optimal transportation with quadraticcost between a empirical measure and a general target probability on $R^d$, with
$d \geq 1$. We provide new results on the uniqueness and stability of the associated optimal transportation potentials, namely,the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.