Rigid and flexible Wasserstein spaces

TL;DR

This paper investigates the isometry groups of p-Wasserstein spaces, constructing examples with mass-splitting isometries and embedding spaces into rigid Wasserstein spaces to understand their geometric properties.

Contribution

It introduces new embeddings of metric spaces into Wasserstein spaces with specific isometry properties, including mass-splitting and rigidity.

Findings

Constructed metric spaces with mass-splitting isometries in Wasserstein spaces.

Embedded any complete, separable metric space into a rigid Wasserstein space.

Demonstrated the existence of isometric embeddings with distinct geometric properties.

Abstract

In this paper, we study isometries of $p$-Wasserstein spaces. In our first result, for every complete and separable metric space $X$ and for every $p\geq1$, we construct a metric space $Y$ such that $X$ embeds isometrically into $Y$, and the $p$-Wasserstein space over $Y$ admits mass-splitting isometries. Our second result is about embeddings into rigid constructions. We show that any complete and separable metric space $X$ can be embedded isometrically into a metric space $Y$ such that the $1$-Wasserstein space is isometrically rigid.