Benamou-Brenier and duality formulas for the entropic cost on $RC{D}^{\ast }(K,N)$ spaces

In this paper we prove that, within the framework of $RC{D}^{\ast }(K,N)$ spaces with $N<\mathrm{\infty }$, 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.