Transportation on spheres via an entropy formula

Blower, Gordon (2023) Transportation on spheres via an entropy formula. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 153 (5). pp. 1467-1478. ISSN 0308-2105

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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 $\mu$ be a probability measure on the sphere ${\bf S}^n$ of the form $d\mu =e^{-U(x)}{\rm d}x$ where ${\rm d}x$ is the rotation invariant probability measure, and $(n-1)I+{\hbox {Hess}}\,U\geq {\kappa _U}I$, where $\kappa _U>0$. Then any probability measure $\nu$ of finite relative entropy with respect to $\mu$ satisfies ${\hbox {Ent}}(\nu \mid \mu ) \geq (\kappa _U/2)W_2(\nu,\, \mu )^2$. 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.

Item Type:
Journal Article
Journal or Publication Title:
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Uncontrolled Keywords:
?? sserstein metriccurvature,transportconvexitymathematics(all) ??
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Deposited On:
20 Jan 2023 17:15
Last Modified:
06 Apr 2024 23:48