Rigid and flexible Wasserstein spaces

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 $p\ge 1$, 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.