Inserted: 1 mar 2012
Last Updated: 16 jan 2013
Journal: Discrete Contin. Dyn. Syst.
Pages: 3043 - 3056
In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition $CD^*(K,N)$ we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of $CD^*(K,N)$ also for intermediate times and in addition the measures along these geodesics have an upper-bound on their densities. This upper-bound depends on the bounds for the densities of the end-point measures, the lower-bound K for the Ricci-curvature, the upper-bound N for the dimension, and on the diameter of the union of the supports of the end-point measures.
Keywords: Ricci curvature, metric measure spaces