Transportation Cost Inequality on Path Spaces with Uniform Distance

TL;DR

This paper establishes transportation-cost inequalities for diffusion processes on path spaces over manifolds, under certain curvature and growth conditions, extending the understanding of probabilistic inequalities in infinite-dimensional settings.

Contribution

It introduces new transportation-cost inequalities for diffusion processes on path spaces over manifolds with specific curvature and growth conditions.

Findings

Transportation-cost inequality holds under bounded Ricci curvature minus gradient of Z.

Condition on Z's linear growth is shown to be optimal.

Inequality applies to path spaces over complete Riemannian manifolds.

Abstract

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.