Self-intersection of optimal geodesics

Let be a geodesic metric measure space. Consider a geodesic in the -Wasserstein space. Then as goes to, the support of and the support of have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of and, we consider the set of times for which this geodesic belongs to the support of. We prove that is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying. The non-branching property is not needed. © 2014 London Mathematical Society.