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

Abstract

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

arXiv admin note: text overlap with arXiv:1802.02463