Isometric study of Wasserstein spaces --- the real line

Abstract

Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space ${\mathcal{W}}_2\left({\mathbb{R}}^n\right)$. It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute $\mathrm{Isom}\left({\mathcal{W}}_p(\mathbb{R})\right)$, the isometry group of the Wasserstein space ${\mathcal{W}}_p(\mathbb{R})$ for all $p\in [1,\infty )\setminus \{2\}$. We show that ${\mathcal{W}}_2(\mathbb{R})$ is also exceptional regarding the parameter $p$: ${\mathcal{W}}_p(\mathbb{R})$ is isometrically rigid if and only if $p\ne 2$. Regarding the underlying space, we prove that the exceptionality of $p=2$ disappears if we replace $\mathbb{R}$ by the compact interval $[0,1]$. Surprisingly, in that case, ${\mathcal{W}}_p\left([0,1]\right)$ is isometrically rigid if and only if $p\ne 1$. Moreover, ${\mathcal{W}}_1\left([0,1]\right)$ admits isometries that split mass, and $\mathrm{Isom}\left({\mathcal{W}}_1\left([0,1]\right)\right)$ cannot be embedded into $\mathrm{Isom}\left({\mathcal{W}}_1(\mathbb{R})\right).$

32 pages, 7 figures. Accepted for publication in Trans. Amer. Math. Soc