The author considers the heat kernel measure $ν$ and the pinned Wiener measure $μ_0$ on the loop group $\Cal L_eG$ over a compact Lie group $G$. It is shown that, for each $p>1$, there exists a unique measurable map $T_p:\Cal L_eG\to \Cal L_eG$ pushing $ν$ forward to $μ_0$ and attaining the Wasserstein distance between these measures. The cost function appearing in the Wasserstein distance is $d_{L^2}(l_1,l_2)^p$, $l_1,l_2\in \Cal L_eG$, where $$ d_{L^2}(l_1,l_2)=\left\{ \int\limits_0^1 ρ(l_1(θ),l_2(θ))^2\,dθ\right\}^{1/2}, $$ and $ρ$ is the Riemannian metric on $G$.
Reviewer: Anatoly N. Kochubei (Kyïv)