Decomposition of geodesics in the Wasserstein space and the globalization property

Understand

We will prove a decomposition for Wasserstein geodesics in the following sense: let $(X,d,m)$ be a non-branching metric measure space verifying $\mathsf{CD}_{loc}(K,N)$ or equivalently $\mathsf{CD}^{*}(K,N)$.

• We prove that every geodesic $\mu_{t}$ in the $L^{2}$-Wasserstein space, with $\mu_{t} \ll m$, is decomposable as the product of two densities, one corresponding to a geodesic with support of codimension one verifying $\mathsf{CD}^{*}(K,N-1)$, and the other associated with a precise one dimensional measure, provided the length map enjoys local Lipschitz regularity.
• The motivation for our decomposition is in the use of the component evolving like $\mathsf{CD}^{*}$ in the globalization problem.
• For a particular class of optimal transportation we prove the linearity in time of the other component, obtaining therefore the global $\mathsf{CD}(K,N)$ for $\mu_{t}$.