Isometric study of Wasserstein spaces – the real line

Abstract:

Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $\mathcal {W}_2(\mathbb {R}^n)$. 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 $\mathrm {Isom}(\mathcal {W}_1(\mathbb {R}))$.