Benamou–Brenier and duality formulas for the entropic cost on $${\textsf {RCD}}^*(K,N)$$RCD∗(K,N) spaces

Abstract

In this paper we prove that, within the framework of RCD * (K, N ) spaces with N < ∞, 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.

An entropic interpolation proof of the HWI inequality

“…shown in [5,9] for Setting 1-(a) and in [14] for rather general metric measure spaces including Setting 1-(b). By the very definition of ϑ ε t and since ε log ρ ε t = ϕ ε t + ψ ε t , this implies…”

“…Namely: -all the regularity and integrability results concerning Schrödinger potentials and entropic interpolations mentioned in Section 2 as well as the dynamic representation of the entropic cost; -the regularizing and contraction properties of (T t ); -the existence of 'good' cut-off functions; -the Benamou-Brenier formula and the Bochner-Lichnerowicz-Weitzenböck inequality. The reader is addressed to [13], [14] for the first point and to [10] for all the others.…”

“…Next lemma provides related regularity and growth controls. Its proof is strongly inspired by [12], [13], [14] and [19]; we thus address the reader to these articles for more details.…”

An Optimal Transport Approach for the Schrödinger Bridge Problem and Convergence of Sinkhorn Algorithm

“…[41,45]), optimal transport theory (e.g. [5,16,18,20,21,32,44,51,54,55,58]), data sciences (e.g. [34,39,40,56,57,62,63] see also the book [28] and references therein).…”

Small Noise Limit and Convexity for Generalized Incompressible Flows, Schrödinger Problems, and Optimal Transport

“…Our dynamical framework is however equivalent, as observed in numerous papers -see for instance [20, Section IV], [26,Cor. 5.8] or the introduction of [27].…”