A geometric study of Wasserstein spaces: Euclidean 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.

26 pages; v3: a few fixes, including correction of Theorem 1.2 (the isometry group of the Wasserstein space of a Euclidean space is in fact only a semi-direct product)

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