Optimal transport maps for Monge–Kantorovich problem on loop groups

Let G be a compact Lie group, and consider the loop group LeG:={ℓ∈C([0,1],G); ℓ(0)=ℓ(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 which pushes ν forward to Fν.