Benamou–Brenier and duality formulas for the entropic cost on RCD∗(K, N) spaces

In this paper we prove that, within the framework of RCD*(K, N) spaces with N < infinity, the entropic cost (i.e. the minimal value of the Schrodinger problem) admits:A threefold dynamical variational representation, in the spirit of the Benamou-Brenier formula for the Wasserstein distance;A Hamilton-Jacobi-Bellman dual representation, in line with Bobkov-Gentil-Ledoux and Otto-Villani results on the duality between Hamilton-Jacobi and continuity equation for optimal transport;A Kantorovich-type duality formula, where the Hopf-Lax semigroup is replaced by a suitable 'entropic' counterpart.We thus provide a complete and unifying picture of the equivalent variational representations of the Schrodinger problem as well as a perfect parallelism with the analogous formulas for the Wasserstein distance. Riemannian manifolds with Ricci curvature bounded from below are a relevant class of RCD* (K, N) spaces and our results are new even in this setting.