A geometric study of Wasserstein spaces: Euclidean spaces

TL;DR

This paper investigates the geometric properties of Wasserstein spaces over Euclidean spaces, revealing unique isometric flows in one dimension and characterizing symmetries and curvature in higher dimensions.

Contribution

It provides a detailed analysis of the isometry groups and geometric structure of Wasserstein spaces over Euclidean spaces, including the discovery of an exotic flow in one dimension.

Findings

Unique isometric flow in the Wasserstein space of the line

All isometries in higher dimensions preserve measure shape

Analysis of curvature and rank of Wasserstein spaces

Abstract

We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) "exotic" isometric flow. This contrasts with the case of higher-dimensional Euclidean spaces, where all isometries of the Wasserstein space preserve the shape of measures. We also study the curvature and various ranks of these spaces.