Let G be a connected Lie group with the Lie algebra ￡: The action of Cameron–Martin space H(￡) on the path space Pe(G) introduced by L. Gross (Illinois J. Math. 36 (1992) 447) is free. Using this fact, we define the H-distance on Pe(G); which enables us to establish a transportation cost inequality on Pe(G): This method will be generalized to the path space over the loop group ￡e(G), so that we obtain a transportation cost inequality for heat measures on ￡e(G).

Dans cette Note, nous allons considérer la distance riemannienne sur le groupe des lacets, qui sera identifie à celle introduite par Hino et Ramirez[M. Hino, J.A. Ramirez, Small-time Gaussian behavior of symmetric diffusion semigroups, Ann. Probab. 31 (2003) 1254–1295]. Une inégalité de transport est établie./ In this Note, we shall consider the Riemannian distance on loop groups, which will be identified to one introduced by Hino and Ramirez[M. Hino, J.A. Ramirez, Small-time Gaussian behavior of symmetric diffusion semigroups, Ann. Probab. 31 (2003) 1254–1295]. A transportation cost inequality is established.

Let (X, ρ) be a Polish space endowed with a probability measure μ. Assume that we can do Malliavin Calculus on (X,μ).Let d :X × X→[0,+∞] be a pseudo-distance. Consider QtF(x) = infy∈X{F(y) + d2(x, y)/2t}. We shall prove that QtF satisfies the Hamilton–Jacobi inequality under suitable conditions. This result will be applied to establish transportation cost inequalities on path groups and loop groups in the spirit of Bobkov, Gentil and Ledoux.

Let G be a compact Lie group, and consider the loop group LeG := {?∈ C([0, 1],G); ? (0) = ? (1) = e}.Let ν be the heat kernel measure at the time 1. For any density function F on LeG such that Entν(F) <∞,we shall prove that there exists a unique optimal transportation map T: LeG → LeG which pushes ν forward to Fν.