Good geodesics satisfying the timelike curvature-dimension condition

Let (M, d, m, MUCH LESS-THAN, <=, tau) be a causally closed, K-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condi-tion wTCDep(K, N) in the sense of Cavalletti and Mondino. We prove the existence of geodesics of probability measures on M which satisfy the entropic semiconvexity inequality defining wTCDep(K, N) and whose densities with respect to m are additionally uniformly L infinity in time. This holds apart from any nonbranching assumption. We also discuss similar results under the timelike measure-contraction property.(c) 2022 Elsevier Ltd. All rights reserved.