Sharpness of Transport Bounds on R\'enyi Divergence on Curved Spaces

Wang's Harnack inequality bounds the R\'enyi divergence between two laws evolved by a diffusion on a manifold with a curvature lower bound. Through the shifted-composition method of Altschuler and Chewi, this bound becomes a transport cost over couplings of the initial laws. We ask when the transport bound gives the true time at which the divergence becomes finite. We prove a lower bound on the transport cost that holds for every coupling on any metric space. For Gaussian initial laws under Ornstein--Uhlenbeck flows the lower bound is attained, and the answer is a phase transition at the variance ratio $\alpha/(\alpha-1)$; the phase diagram does not depend on the curvature. We extend the result to non-commuting covariances in $\R^d$ and to heat kernels on hyperbolic space, where negative curvature makes the bound strictly loose.