The existence of geodesics in Wasserstein spaces over path groups and loop groups

Loop group is an important curved infinite dimensional space, and the stochastic analysis theory on it relies heavily on two probability measures: pinned Wiener measure and heat kernel measure. This work solves the Monge-Kantorovich problem over loop group, and shows that there exists a unique reversible optimal transport map between pinned Wiener measure and heat kernel measure. Based on this result, the geometry of the Wasserstein space over loop group is developed. In particular, it proves that this Wasserstein space is a geodesic space. This fact is the foundation and key point to develop Lott-Villani-Sturm Ricci curvature over loop group.