Analysis on free Riemannian path spaces
Abstract The gradient operator is defined on the free path space with reference measure P μ , the law of the Brownian motion on the base manifold with initial distribution μ, where μ has strictly positive density w.r.t. the volume measure. The formula of integration by parts is established for the underlying directional derivatives, which implies the closability of the gradient operator so that it induces a conservative Dirichlet form on the free path space. The log-Sobolev inequality for this Dirichlet form is established and, consequently, the transportation cost inequality is obtained for the associated intrinsic distance.
Transportation-cost inequality on path spaces with uniform distance
Abstract Let M be a complete Riemannian manifold and μ the distribution of the diffusion process generated by 1 2 ( Δ + Z ) where Z is a C 1 -vector field. When Ric − ∇ Z is bounded below and Z has, for instance, linear growth, the transportation-cost inequality with respect to the uniform distance is established for μ on the path space over M . A simple example is given to show the optimality of the condition.