Self-Intersection of Optimal geodesics

Abstract

Let $(X,d,m)$ be a geodesic metric measure space. Consider a geodesic $\mu_{t}$ in the $L^{2}$-Wasserstein space. Then as $s$ goes to $t$ the support of $\mu_{s}$ and the support of $\mu_{t}$ have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. We consider for each $t$ the set of times for which a geodesic belongs to the support of $\mu_{t}$ and we prove that $t$ is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying $\mathsf{CD}(K,\infty)$. The non branching property is not needed.