Transportation-cost inequality on path spaces with uniform distance

Let M be a complete Riemannian manifold and [mu] the distribution of the diffusion process generated by where Z is a C1-vector field. When is bounded below and Z has, for instance, linear growth, the transportation-cost inequality with respect to the uniform distance is established for [mu] on the path space over M. A simple example is given to show the optimality of the condition.Transportation-cost inequality Path space Damped gradient Quasi-invariant flow Uniform distance