Some inequalities on Riemannian manifolds linking Entropy, Fisher information, Stein discrepancy and Wasserstein distance

摘要

For a complete connected Riemannian manifold M let V is an element of C2(M) be such that mu(dx) = e-V(x)vol(dx) is a probabil-ity measure on M. Taking mu as reference measure, we derive inequalities for probability measures on M linking relative entropy, Fisher information, Stein discrepancy and Wasser-stein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds.(c) 2023 Elsevier Inc. All rights reserved.