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 $\mathrm{Isom}\left(\mathcal{W}_p(\mathbb{R})\right)$, 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 $p\neq 2$. 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 $p\neq1$. 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 $\mathrm{Isom}\left(\mathcal{W}_1(\mathbb{R})\right).$