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(\mathbb{R}^n)$. 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}(\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\}$. 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([0,1])$ is isometrically rigid if and only if $ p\neq 1$. Moreover, $ \mathcal {W}_1([0,1])$ admits isometries that split mass, and $ \mathrm {Isom}(\mathcal {W}_1([0,1]))$ cannot be embedded into $ \mathrm {Isom}(\mathcal {W}_1(\mathbb{R}))$