Calculus of Variations and Geometric Measure Theory

N. De Ponti - A. Mondino

Entropy-Transport distances between unbalanced metric measure spaces

created by mondino on 22 Sep 2020
modified by deponti on 24 Jul 2022


Accepted Paper

Inserted: 22 sep 2020
Last Updated: 24 jul 2022

Journal: Probability Theory and Related Fields
Year: 2020


Inspired by the recent theory of Entropy-Transport problems and by the $\mathbf{D}$-distance of Sturm on \emph{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.