0 A ug 2 00 9 Transportation - Cost Inequalities on Path Space Over Manifolds with Boundary ∗

Let L = ∆+Z for a C1 vector field Z on a complete Riemannian manifold possibly with a boundary. By using the uniform distance, a number of transportation-cost inequalities on the path space for the (reflecting) L-diffusion process are proved to be equivalent to the curvature condition Ric−∇Z ≥ −K and the convexity of the boundary (if exists). These inequalities are new even for manifolds without boundary, and are partly extended to non-convex manifolds by using a conformal change of metric which makes the boundary from non-convex to convex. AMS subject Classification: 60J60, 58G60.