Transportation Cost Inequality on Path Spaces with Uniform Distance

Starting from a sequence of independent Wright-Fisher diffusion processes on $[0,1]$, we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in Let$M$be a complete Riemnnian manifold and$\mu$the distribution of the diffusion process generated by$\ff 1 2\DD+Z$where$Z$is a$C^1$-vector field. When$\Ric-\nn Z$is bounded below and$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.