Optimal transport maps for Monge–Kantorovich problem on loop groups

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. © 2006 Elsevier Inc. All rights reserved