Isometric study of Wasserstein spaces - the real line

Gehér, G. P., Titkos, T. and Virosztek, D. (2020) Isometric study of Wasserstein spaces - the real line. Transactions of the American Mathematical Society, 373 (8). pp. 5855-5883. ISSN 1088-6850 doi: 10.1090/tran/8113

Abstract/Summary

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}))$.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/85689
Identification Number/DOI	10.1090/tran/8113
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	American Mathematical Society
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