## Isometric study of Wasserstein spaces - the real line
Gehér, G. P., Titkos, T. and Virosztek, D.
(2020)
It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. To link to this item DOI: 10.1090/tran/8113 ## Abstract/SummaryRecently 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}))$.
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