Benamou-Brenier and duality formulas for the entropic cost on $RCD^*(K,N)$ spaces

TL;DR

This paper establishes multiple variational and duality formulas for the entropic cost in $RCD^*(K,N)$ spaces, unifying the Schrödinger problem with optimal transport theory.

Contribution

It introduces a comprehensive set of variational representations and duality formulas for the entropic cost on $RCD^*(K,N)$ spaces, extending known results to this framework.

Findings

Provides a threefold dynamical variational representation.

Establishes a Hamilton-Jacobi-Bellman duality formula.

Derives a Kantorovich-type duality with an entropic semigroup.

Abstract

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.