Wasserstein geometry of Gaussian measures

Abstract

This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the $L^{2}$-Wasserstein geometry. The space of Gaussian measures is of finite dimension, which allows to write down the explicit Riemannian metric which in turn induces the $L^{2}$-Wasserstein distance. Moreover, its completion as a metric space provides a complete picture of the singular behavior of the $L^{2}$-Wasserstein geometry. In particular, the singular set is stratified according to the dimension of the support of the Gaussian measures, providing an explicit nontrivial example of Alexandrov space with extremal sets.