Absolute continuity of Wasserstein barycenters on manifolds with a lower Ricci curvature bound
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911300817256448 |
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| author | Ma, Jianyu |
| author_facet | Ma, Jianyu |
| contents | Given a complete Riemannian manifold $M$ with a lower Ricci curvature bound, we consider barycenters in the Wasserstein space $\mathcal{W}_2(M)$ of probability measures on $M$. We refer to them as Wasserstein barycenters, which by definition are probability measures on $M$. The goal of this article is to present a novel approach to proving their absolute continuity. We introduce a new class of displacement functionals exploiting the Hessian equality for Wasserstein barycenters. To provide suitable instances of such functionals, we revisit Souslin space theory, Dunford-Pettis theorem and the de la Vallée Poussin criterion for uniform integrability. Our method shows that if a probability measure $\mathbb{P}$ on $\mathcal{W}_2(M)$ gives mass to absolutely continuous measures on $M$, then its unique barycenter is also absolutely continuous. This generalizes the previous results on compact manifolds by Kim and Pass arXiv:1412.7726 [math.AP] . |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_13832 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Absolute continuity of Wasserstein barycenters on manifolds with a lower Ricci curvature bound Ma, Jianyu Probability Analysis of PDEs Differential Geometry 49Q22, 51K10, 28A05 Given a complete Riemannian manifold $M$ with a lower Ricci curvature bound, we consider barycenters in the Wasserstein space $\mathcal{W}_2(M)$ of probability measures on $M$. We refer to them as Wasserstein barycenters, which by definition are probability measures on $M$. The goal of this article is to present a novel approach to proving their absolute continuity. We introduce a new class of displacement functionals exploiting the Hessian equality for Wasserstein barycenters. To provide suitable instances of such functionals, we revisit Souslin space theory, Dunford-Pettis theorem and the de la Vallée Poussin criterion for uniform integrability. Our method shows that if a probability measure $\mathbb{P}$ on $\mathcal{W}_2(M)$ gives mass to absolutely continuous measures on $M$, then its unique barycenter is also absolutely continuous. This generalizes the previous results on compact manifolds by Kim and Pass arXiv:1412.7726 [math.AP] . |
| title | Absolute continuity of Wasserstein barycenters on manifolds with a lower Ricci curvature bound |
| topic | Probability Analysis of PDEs Differential Geometry 49Q22, 51K10, 28A05 |
| url | https://arxiv.org/abs/2310.13832 |