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

摘要

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD~*(K,N) we always have geodesies 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 geodesies 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.