Self-Intersection of Optimal geodesics

TL;DR

This paper proves a precise self-intersection property of optimal geodesics in Wasserstein space on (K,) spaces, showing that geodesics' supports overlap in a Lebesgue density sense without requiring non-branching assumptions.

Contribution

It establishes a detailed self-intersection property of optimal geodesics in Wasserstein space under (K,) conditions, without assuming non-branching.

Findings

Supports of geodesics overlap in Lebesgue density 1.

The result applies to (K,) spaces with bounded densities.

Non-branching property is not required.

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.