Entropy-Transport distances between unbalanced metric measure spaces

TL;DR

This paper introduces a new class of distances between metric measure spaces with possibly different total mass, based on entropy-transport problems, and explores their properties, examples, and applications to spaces with Ricci curvature bounds.

Contribution

It defines and analyzes a novel family of complete, separable distances between unbalanced metric measure spaces, extending existing concepts like Sturm's D-distance and incorporating entropic and transport-based metrics.

Findings

Introduces a new class of distances between unbalanced metric measure spaces.

Provides explicit examples including Hellinger-Kantorovich based distances.

Establishes compactness and stability results for spaces with Ricci curvature lower bounds.

Abstract

Inspired by the recent theory of Entropy-Transport problems and by the $\mathbf{D}$-distance of Sturm on normalised metric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly different total mass. We provide several explicit examples of such distances, where a prominent role is played by a geodesic metric based on the Hellinger-Kantorovich distance. Moreover, we discuss some limiting cases of the theory, recovering the "pure transport" $\mathbf{D}$-distance and introducing a new class of "pure entropic" distances. We also study in detail the topology induced by such Entropy-Transport metrics, showing some compactness and stability results for metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense.