Sharpness of Transport Bounds on Renyi Divergence along Diffusions

Two initial laws are moved by the same diffusion, whose Bakry--\'Emery curvature is at least $\kappa$. Examples are the forward process of a diffusion model, which is an Ornstein--Uhlenbeck flow, and the Langevin diffusion used in sampling. The shifted-composition method of Altschuler and Chewi bounds the R\'enyi divergence between the two evolved laws by a transport cost over couplings of the initial laws, and so gives a time after which the divergence is finite. We ask when this time is the true one. We prove a lower bound on the transport cost of every coupling on any Polish metric space, and a tail argument covers the remaining case. For Gaussian initial laws under Ornstein--Uhlenbeck flows this gives the threshold in closed form, and the answer is a phase transition at the ratio $\alpha/(\alpha-1)$ of the standard deviations, the same for every $\kappa$. At and above the transition the bound gives the true time. Below it, except at equal variances, the bound misses the true time whenever that time is finite, by an explicit gap that no coupling can remove. We extend the result to non-commuting covariances in $\R^d$ and compare it with a companion bound, which gives the true time when the ratio is at most one.