The isometry group of Wasserstein spaces: the Hilbertian case
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866909289153560576 |
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| author | Gehér, György Pál Titkos, Tamás Virosztek, Dániel |
| author_facet | Gehér, György Pál Titkos, Tamás Virosztek, Dániel |
| contents | Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{R}^n\right)$, we describe the isometry group $\mathrm{Isom}\left(\mathcal{W}_p (E)\right)$ for all parameters $0 < p < \infty$ and for all separable real Hilbert spaces $E.$ In particular, we show that $\mathcal{W}_p(X)$ is isometrically rigid for all Polish space $X$ whenever $0<p<1$. This is a consequence of our more general result: we prove that $\mathcal{W}_1(X)$ is isometrically rigid if $X$ is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters $p>1$, by solving Kloeckner's problem affirmatively on the existence of mass-splitting isometries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_02037 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The isometry group of Wasserstein spaces: the Hilbertian case Gehér, György Pál Titkos, Tamás Virosztek, Dániel Metric Geometry Mathematical Physics Functional Analysis Probability Primary: 54E40, 46E27, Secondary: 60A10, 60B05 Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{R}^n\right)$, we describe the isometry group $\mathrm{Isom}\left(\mathcal{W}_p (E)\right)$ for all parameters $0 < p < \infty$ and for all separable real Hilbert spaces $E.$ In particular, we show that $\mathcal{W}_p(X)$ is isometrically rigid for all Polish space $X$ whenever $0<p<1$. This is a consequence of our more general result: we prove that $\mathcal{W}_1(X)$ is isometrically rigid if $X$ is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters $p>1$, by solving Kloeckner's problem affirmatively on the existence of mass-splitting isometries. |
| title | The isometry group of Wasserstein spaces: the Hilbertian case |
| topic | Metric Geometry Mathematical Physics Functional Analysis Probability Primary: 54E40, 46E27, Secondary: 60A10, 60B05 |
| url | https://arxiv.org/abs/2102.02037 |