Transportation cost inequalities on path spaces over Riemannian manifolds
Transportation cost inequalities of the form \(W_2(f\mu,\mu)^2 \leq 2\mu (f\log f)\), where \(W_2\) is the Wasserstein distance and \(f\) is a nonnegative function such that \(\mu (f)=1\), have been obtained by \textit{M. Talagrand} [Geom. Funct. Anal. 6, 587-600 (1996; Zbl 0859.46030)] for the standard Gaussian measure \(\mu\) on \({\mathbb R}^d\), and extended by several authors to measures \(\mu\) on manifolds and on path spaces over \({\mathbb R}^d\), provided a logarithmic Sobolev inequality holds under \(\mu\). In this paper a transportation cost inequality is established on the path space over a connected complete Riemannian manifold with Ricci curvature bounded from below, using the natural distance on path space defined as the \(L^2\) norm of the Riemannian distance along paths.
scientific article; zbMATH DE number 1952931