IMPROVED GEODESICS FOR THE REDUCED CURVATURE-DIMENSION CONDITION IN BRANCHING METRIC SPACES

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 con- vexity 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 mea- sures, 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.