Isometric study of Wasserstein spaces --- the real line

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 $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}({\mathcal{W}}_p(\mathbb{R}))$, the isometry group of the Wasserstein space ${\mathcal{W}}_p(\mathbb{R})$ for all $p \in [1, \infty)\setminus{2}$\mathrm{.}Weshowthat$$\mathcal{W}$_2($\mathbb{R}$)$ is also exceptional regarding the parameter $p$: ${\mathcal{W}}_p(\mathbb{R})$ is isometrically rigid if and only if $p\mathrm{\ne }2$. 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, ${\mathcal{W}}_p([0,1])$ is isometrically rigid if and only if $p\mathrm{\ne }1$. Moreover, ${\mathcal{W}}_1([0,1])$ admits isometries that split mass, and $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}\left({\mathcal{W}}_1([0,1])\right)$ cannot be embedded into $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}({\mathcal{W}}_1(\mathbb{R}))\mathrm{.}$