Generalized curvature for the optimal transport problem induced by a Tonelli Lagrangian

TL;DR

This paper introduces a generalized curvature concept motivated by optimal transport with Tonelli Lagrangians, linking curvature non-negativity to displacement convexity of entropy in Wasserstein spaces.

Contribution

It defines a new generalized curvature for optimal transport problems induced by Tonelli Lagrangians and establishes its connection to displacement convexity of entropy.

Findings

Non-negativity of the generalized curvature implies displacement convexity.

The generalized curvature is motivated by optimal transport with Tonelli Lagrangians.

The results connect curvature properties to entropy convexity in Wasserstein spaces.

Abstract

We propose a generalized curvature that is motivated by the optimal transport problem on $\mathbb{R}^d$ with cost induced by a Tonelli Lagrangian $L$. We show that non-negativity of the generalized curvature implies displacement convexity of the generalized entropy functional on the $L-$Wasserstein space along $C^2$ displacement interpolants.