Probability distance inequalities on Riemannian manifolds and path spaces

We construct Otto-Villani's coupling for general reversible diffusion processes on a Riemannian manifold. As an application, some new estimates are obtained for Wasserstein distances by using a Sobolev-Poincaré type inequality introduced by Lata?a and Oleszkiewicz. The corresponding concentration estimates of the measure are presented. Finally, our main result is applied to obtain the transportation cost inequalities on the path space with respect to both of the L²-distance and the intrinsic distance. In particular, Talagrand's inequality holds on the path space over a compact manifold.