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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Isometric study of Wasserstein spaces – the real line


Authors: György Pál Gehér, Tamás Titkos and Dániel Virosztek
Journal: Trans. Amer. Math. Soc. 373 (2020), 5855-5883
MSC (2010): Primary 54E40, 46E27; Secondary 60A10, 60B05
DOI: https://doi.org/10.1090/tran/8113
Published electronically: May 26, 2020
MathSciNet review: 4127894
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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 $\mathrm {Isom}(\mathcal {W}_1(\mathbb {R}))$.


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Additional Information

György Pál Gehér
Affiliation: Department of Mathematics and Statistics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, United Kingdom
Email: g.p.geher@reading.ac.uk, gehergyuri@gmail.com

Tamás Titkos
Affiliation: Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15 Budapest H-1053, Hungary; and BBS University of Applied Sciences, Alkotmány u. 9 Budapest H-1054, Hungary
Email: titkos@renyi.hu

Dániel Virosztek
Affiliation: Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
Email: daniel.virosztek@ist.ac.at

Keywords: Wasserstein space, isometric embeddings, isometric rigidity, exotic isometry flow
Received by editor(s): August 19, 2019
Received by editor(s) in revised form: January 20, 2020
Published electronically: May 26, 2020
Additional Notes: The first author was supported by the Leverhulme Trust Early Career Fellowship (ECF-2018-125) and also by the Hungarian National Research, Development and Innovation Office - NKFIH (grant no. K115383).
The second author was supported by the Hungarian National Research, Development and Innovation Office - NKFIH (grant no. PD128374 and grant no. K115383), by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the ÚNKP-18-4-BGE-3 New National Excellence Program of the Ministry of Human Capacities.
The third author was supported by the ISTFELLOW program of the Institute of Science and Technology Austria (project code IC1027FELL01), by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office - NKFIH (grants no. K124152 and KH129601).
Article copyright: © Copyright 2020 American Mathematical Society