  In this paper we prove that, within the framework of $RCD^*(K,N)$ spaces with
$N < \infty$, the entropic cost (i.e. the minimal value of the Schr\"odinger
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 Schr\"odinger problem (still missing even in the
Riemannian setting) as well as a perfect parallelism with the analogous
formulas for the Wasserstein distance.
