From the heat measure to the pinned Wiener measure on loop groups (Q645939)

The author considers the heat kernel measure \(\nu\) and the pinned Wiener measure \(\mu_0\) on the loop group \(\mathcal L_eG\) over a compact Lie group \(G\). It is shown that, for each \(p>1\), there exists a unique measurable map \(T_p:\mathcal L_eG\to \mathcal L_eG\) pushing \(\nu\) forward to \(\mu_0\) and attaining the Wasserstein distance between these measures. The cost function appearing in the Wasserstein distance is \(d_{L^2}(l_1,l_2)^p\), \(l_1,l_2\in \mathcal L_eG\), where \[ d_{L^2}(l_1,l_2)=\left\{ \int\limits_0^1 \rho (l_1(\theta ),l_2(\theta ))^2\,d\theta \right\}^{1/2}, \] and \(\rho\) is the Riemannian metric on \(G\).