Decomposition of geodesics in the Wasserstein space and the globalization property
Let $(X,d,m)$ be a non-branching metric measure space verifying $\mathsfCD_loc(K,N)$ or equivalently $\mathsfCD^*(K,N)$. In this note we show that given a geodesic $μ_t$ in the $L^2$-Wasserstein space, it is always possible to write the density of $μ_t$ as the product of two densities, one corresponding to a geodesic with support of codimension one verifying $\mathsfCD(K,N-1)$, and the other associated with a one dimensional measure. For a particular class of optimal transportation we prove the linearity in time of the other component, obtaining therefore the full $\mathsfCD(K,N)$ for $μ_t$. This result can be therefore interpret as the "self-improving property" for $\mathsfCD^*(K,N)$ or as a partial globalization theorem for $\mathsfCD(K,N)$. In the setting of infinitesimally strictly convex metric measure space, we also write explicitly the one dimensional density obtaining a complete and explicit decomposition of the density.