Displacement interpolations from a Hamiltonian point of view

TL;DR

This paper presents a Hamiltonian systems perspective on displacement interpolations in optimal transport, unifying various curvature-related results across different geometric settings.

Contribution

It introduces a Hamiltonian framework that unifies and generalizes existing results on displacement interpolations and curvature bounds in Riemannian and Finsler geometries.

Findings

Provides a Hamiltonian approach to displacement interpolations.

Unifies curvature bounds in different geometric contexts.

Extends results to manifolds with Ricci flow background.

Abstract

One of the most well-known results in the theory of optimal transportation is the equivalence between the convexity of the entropy functional with respect to the Riemannian Wasserstein metric and the Ricci curvature lower bound of the underlying Riemannian manifold. There are also generalizations of this result to the Finsler manifolds and manifolds with a Ricci flow background. In this paper, we study displacement interpolations from the point of view of Hamiltonian systems and give a unifying approach to the above mentioned results.