Transportation cost inequalities on path and loop groups (Q1763935)

The transportation cost inequalities compare the square of the Wasserstein distance between a measure \(\mu\) on a metric space \(X\) and the measure \(f\mu\), where \(f\) is a positive continuous function, with the corresponding Kullback information; see \textit{H. Djellout, A. Guillin} and \textit{L. Wu} [Ann. Probab. 32, No. 3B, 2702--2732 (2004; Zbl 1061.60011)] for a brief survey of results in this direction for various classes of spaces and measures. The authors consider the case where \(X\) is the path space on a connected Lie group, and \(\mu\) is the appropriate generalization of the Wiener measure. They introduce and study a distance on \(X\) and prove the corresponding transportation cost inequality. The case, where \(X\) is the path space on the loop group over a Lie group, is also considered.