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

Text (Transportationsphere)
Transportationsphere.pdf
Download (223kB)

Text (TransportationsphereR1)
TransportationsphereR1.pdf - Accepted Version
Available under License Creative Commons Attribution.
Download (231kB)

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 $d\mu =e^{-U(x)}{\rm d}x$ 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.