Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities

Under general assumptions on the target distribution p^star, we establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores, with optimal dependence on time and dimension. As applications, we obtain Wasserstein discretization bounds for Euler-type samplers in dimension d: with N discretization steps, the error achieves the optimal rate √d/N up to logarithmic factors. Moreover, the constants do not deteriorate exponentially with the spatial extent of p^star. We also show that the one-sided Lipschitz control yields a globally Lipschitz transport map from the standard Gaussian to p^star, which implies Poincar'e and log-Sobolev inequalities for a broad class of probability measures.