Decomposition of Wasserstein Geodesics in spaces with lower curvature bound

Fabio Cavalletti
RWTH Aachen

Let (X, d,m) be a non-branching metric measure space verifying CDloc(K,N) or equivalently CD*(K,N). We prove that every geodesic µt in the L2-Wasserstein space with µt « m, is decomposable as the product of two densities, one corresponding to a geodesic with support of codimension one verifying CD*(K,N-1), and the other associated with a precise one dimensional measure. For a particular class of optimal transportation we prove the linearity in time of the other component, obtaining the full CD(K,N) for µt and therefore a partial globalization theorem for CD(K,N).

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