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.

References in corpus (2)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Decomposition of geodesics in the Wasserstein space and the globalization property

Cited by in corpus (1)

• Decomposition of geodesics in the Wasserstein space and the globalization property