Isometric rigidity of the Wasserstein space over the plane with the maximum metric

Abstract

We study $p$-Wasserstein spaces over the branching spaces ${\mathbb{R}}^2$ and $[-1,1{]}^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\ge 1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over ${\mathbb{R}}^2$ is not rigid. Also, we highlight that the $1$-Wasserstein space is not rigid over the closed interval $[-1,1]$, while according to our result, its two-dimensional analog, the closed unit ball $[-1,1{]}^2$ with the more complicated geodesic structure is rigid.

27 pages, 7 figures. v2: a gap in the proof of Proposition 3 (in the first version: Proposition 2) has been filled and some other proofs have been streamlined. v3: accepted manuscript version