Transportation-Cost Inequalities on Path Space over Manifolds with Boundary

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.

2010 Mathematics Subject Classification: 60J60, 58G60.

Keywords and Phrases: Transportation-cost inequality, curvature, second fundamental form, path space.

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