$W$-entropy formulas and Langevin deformation on the $L^q$-Wasserstein space over Riemannian manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911018423156736 |
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| author | Lei, Rong Li, Xiang-Dong Wang, Yu-Zhao |
| author_facet | Lei, Rong Li, Xiang-Dong Wang, Yu-Zhao |
| contents | We first prove the $W$-entropy formula and rigidity theorem for the geodesic flow on the $L^q$-Wasserstein space over a complete Riemannian manifold with bounded geometry condition. Then we introduce the Langevin deformation on the $L^q$-Wasserstein space over a complete Riemannian manifold, which interpolates between the $p$-Laplacian heat equation and the geodesic flow on the $L^q$-Wasserstein space, where ${1\over p}+{1\over q}=1$, $1< p, q<\infty$. The local existence, uniqueness and regularity of the Langevin deformation on the $L^q$-Wasserstein space over the Euclidean space and a compact Riemannian manifold are proved for $q\in [2, \infty)$. We further prove the $W$-entropy-information formula and the rigidity theorem for the Langevin deformation on the $L^q$-Wasserstein space over an $n$-dimensional complete Riemannian manifold with non-negative Ricci curvature, where $q\in (1,\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01279 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $W$-entropy formulas and Langevin deformation on the $L^q$-Wasserstein space over Riemannian manifolds Lei, Rong Li, Xiang-Dong Wang, Yu-Zhao Probability Primary 58J35, 58J65, Secondary 35K92, 60H30 We first prove the $W$-entropy formula and rigidity theorem for the geodesic flow on the $L^q$-Wasserstein space over a complete Riemannian manifold with bounded geometry condition. Then we introduce the Langevin deformation on the $L^q$-Wasserstein space over a complete Riemannian manifold, which interpolates between the $p$-Laplacian heat equation and the geodesic flow on the $L^q$-Wasserstein space, where ${1\over p}+{1\over q}=1$, $1< p, q<\infty$. The local existence, uniqueness and regularity of the Langevin deformation on the $L^q$-Wasserstein space over the Euclidean space and a compact Riemannian manifold are proved for $q\in [2, \infty)$. We further prove the $W$-entropy-information formula and the rigidity theorem for the Langevin deformation on the $L^q$-Wasserstein space over an $n$-dimensional complete Riemannian manifold with non-negative Ricci curvature, where $q\in (1,\infty)$. |
| title | $W$-entropy formulas and Langevin deformation on the $L^q$-Wasserstein space over Riemannian manifolds |
| topic | Probability Primary 58J35, 58J65, Secondary 35K92, 60H30 |
| url | https://arxiv.org/abs/2506.01279 |