Decomposition of geodesics in the Wasserstein space and the globalization property

Abstract

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 $\mu_{t}$. The result can be therefore interpret as a globalization theorem for $\mathsf{CD}(K,N)$ for this class of optimal transportation, or as a ``self-improving property'' for $\mathsf{CD}^{*}(K,N)$. Assuming more regularity, namely in the setting of infinitesimally strictly convex metric measure space, the one dimensional density is the product of two differentials giving more insight on the density decomposition.