Some transportation cost inequalities are established on the path space over a connected complete Riemannian manifold with Ricci curvature bounded from below. The reference distance on the path space is the $L^2$-norm of the Riemannian distance along paths.
"Transportation cost inequalities on path spaces over Riemannian manifolds." Illinois J. Math. 46 (4) 1197 - 1206, Winter 2002. https://doi.org/10.1215/ijm/1258138474