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 $p\neq 2$. 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 $\mathrm{Isom}\left(\mathcal{W}_1(\mathbb{R})\right).$
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2020
- DOI:
- 10.48550/arXiv.2002.00859
- arXiv:
- arXiv:2002.00859
- Bibcode:
- 2020arXiv200200859P
- Keywords:
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- Mathematics - Metric Geometry;
- Mathematical Physics;
- Mathematics - Functional Analysis;
- Mathematics - Probability;
- Primary: 54E40;
- 46E27. Secondary: 60A10;
- 60B05
- E-Print:
- 32 pages, 7 figures. Accepted for publication in Trans. Amer. Math. Soc