Transportation-Cost Inequalities on Path Space Over Manifolds with Boundary

TL;DR

This paper establishes new transportation-cost inequalities on path spaces over manifolds with boundary, linking them to curvature conditions and boundary convexity, with extensions to non-convex manifolds via conformal metric changes.

Contribution

It introduces novel transportation-cost inequalities on path spaces that are equivalent to curvature and boundary convexity conditions, including extensions to non-convex manifolds.

Findings

Transportation-cost inequalities are equivalent to curvature bounds and boundary convexity.

New inequalities are established even for manifolds without boundary.

Extensions to non-convex manifolds are achieved through conformal metric changes.

Abstract

Let $L=\DD+Z$ for a $C^1$ 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 $\Ric-\nn Z\ge - K$ 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.