Transportation on spheres via an entropy formula

Blower, Gordon (2022) Transportation on spheres via an entropy formula. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. ISSN 0308-2105

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Abstract

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 $d\mu =e^{-U(x)}dx$ where $dx$ 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 Lichn\'erowicz integral.

Item Type:
Journal Article
Journal or Publication Title:
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600
Subjects:
ID Code:
184521
Deposited By:
Deposited On:
20 Jan 2023 17:15
Refereed?:
Yes
Published?:
Published
Last Modified:
26 Jan 2023 01:58