Transportation on spheres via an entropy formula
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866909322377691136 |
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| author | Blower, Gordon |
| author_facet | Blower, Gordon |
| contents | The paper proves transportation inequalities for probability measures on spheres for the Wasserstein metrics with respect to cost functions that are powers of the geodesic distance. Let $μ$ be a probability measure on the sphere ${\bf S}^n$ of the form $dμ=e^{-U(x)}dx$ where $dx$ is the rotation invariant probability measure, and $(n-1)I+{\hbox{Hess}}\,U\geq {κ_U}I$, where $κ_U>0$. Then any probability measure $ν$ of finite relative entropy with respect to $μ$ satisfies ${\hbox{Ent}}(ν\midμ) \geq (κ_U/2)W_2(ν, μ)^2$. The proof uses an explicit formula for the relative entropy which is also valid on connected and compact $C^\infty$ smooth Riemannian manifolds without boundary. A variation of this entropy formula gives the Lichnérowicz integral. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_06191 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Transportation on spheres via an entropy formula Blower, Gordon Probability 60E15, 58C35 The paper proves transportation inequalities for probability measures on spheres for the Wasserstein metrics with respect to cost functions that are powers of the geodesic distance. Let $μ$ be a probability measure on the sphere ${\bf S}^n$ of the form $dμ=e^{-U(x)}dx$ where $dx$ is the rotation invariant probability measure, and $(n-1)I+{\hbox{Hess}}\,U\geq {κ_U}I$, where $κ_U>0$. Then any probability measure $ν$ of finite relative entropy with respect to $μ$ satisfies ${\hbox{Ent}}(ν\midμ) \geq (κ_U/2)W_2(ν, μ)^2$. The proof uses an explicit formula for the relative entropy which is also valid on connected and compact $C^\infty$ smooth Riemannian manifolds without boundary. A variation of this entropy formula gives the Lichnérowicz integral. |
| title | Transportation on spheres via an entropy formula |
| topic | Probability 60E15, 58C35 |
| url | https://arxiv.org/abs/2207.06191 |