# Isometric study of Wasserstein spaces - the real line

Geher, G., Titkos, T. and Virosztek, D. (2020) Isometric study of Wasserstein spaces - the real line. Transactions of the American Mathematical Society. ISSN 1088-6850 (In Press)

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## Abstract/Summary

Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $\mathcal{W}_2(\R^n)$. 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 $\isom\ler{\Wp(\R)}$, the isometry group of the Wasserstein space $\Wp(\R)$ for all $p \in [1, \infty)\setminus\{2\}$. We show that $\mathcal{W}_2(\R)$ is also exceptional regarding the parameter $p$: $\Wp(\R)$ 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 $\R$ by the compact interval $[0,1]$. Surprisingly, in that case, $\Wp\zo$ is isometrically rigid if and only if $p\neq1$. Moreover, $\Wo\zo$ admits isometries that split mass, and $\isom\ler{\Wo\zo}$ cannot be embedded into $\isom\ler{\wor}$.

Item Type: Article Yes Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics 85689 American Mathematical Society

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