Transportation-cost inequality on path spaces with uniform distance

Let MM be a complete Riemannian manifold and μμ the distribution of the diffusion process generated by 12(Δ+Z) where ZZ is a C1C1-vector field. When Ric−∇Z is bounded below and ZZ has, for instance, linear growth, the transportation-cost inequality with respect to the uniform distance is established for μμ on the path space over MM. A simple example is given to show the optimality of the condition.