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We will prove a decomposition for Wasserstein geodesics in the followingsense: 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 thatevery geodesic $\mu_{t}$ in the $L^{2}$-Wasserstein space, with $\mu_{t} \llm$, is decomposable as the product of two densities, one corresponding to ageodesic with support of codimension one verifying $\mathsf{CD}^{*}(K,N-1)$,and the other associated with a precise one dimensional measure, provided thelength map enjoys local Lipschitz regularity. The motivation for ourdecomposition is in the use of the component evolving like $\mathsf{CD}^{*}$ inthe globalization problem.  For a particular class of optimal transportation we prove the linearity intime of the other component, obtaining therefore the global $\mathsf{CD}(K,N)$for $\mu_{t}$. The result can be therefore interpret as a globalization theoremfor $\mathsf{CD}(K,N)$ for this class of optimal transportation, or as a``self-improving property'' for $\mathsf{CD}^{*}(K,N)$. Assuming moreregularity, namely in the setting of infinitesimally strictly convex metricmeasure space, the one dimensional density is the product of two differentialsgiving more insight on the density decomposition.

Decomposition of geodesics in the Wasserstein space and the globalization property