Optimal transport maps for Monge–Kantorovich problem on loop groups

Abstract

Let G be a compact Lie group, and consider the loop group L e G : = { 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 L e G such that Ent ν ( F ) < ∞ , we shall prove that there exists a unique optimal transportation map T : L e G → L e G which pushes ν forward to Fν.