Transportation-cost inequalities on path space over manifolds with boundary

Abstract

Let $L=\Delta +Z$ for a $C^1$ vector field $Z$ on a complete Riemannian manifold possibly with a boundary. 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 $Ric-\nabla Z\ge -K$ 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.