The Exponential Formula for the Wasserstein Metric

TL;DR

This paper introduces new transport metrics with improved convexity properties to analyze Wasserstein gradient flows, providing a novel proof of the exponential formula and related properties.

Contribution

It develops a class of transport metrics with better convexity and uses them to prove the exponential formula for Wasserstein metric, offering new insights into gradient flow analysis.

Findings

New class of transport metrics with enhanced convexity

A novel proof of the exponential formula for Wasserstein metric

Simplified proofs of gradient flow properties such as contraction and energy dissipation

Abstract

A recurring obstacle in the study of Wasserstein gradient flow is the lack of convexity of the square Wasserstein metric. In this paper, we develop a class of transport metrics that have better convexity properties and use these metrics to prove an Euler-Lagrange equation characterizing Wasserstein discrete gradient flow. We then apply these results to give a new proof of the exponential formula for the Wasserstein metric, mirroring Crandall and Liggett's proof of the corresponding Banach space result. We conclude by using our approach to give simple proofs of properties of the gradient flow, including the contracting semigroup property and energy dissipation inequality.