Inspired by the recent theory of Entropy-Transport problems and by the 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” 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.

De Ponti, N., Mondino, A. (2022). Entropy-Transport distances between unbalanced metric measure spaces. PROBABILITY THEORY AND RELATED FIELDS, 184(1-2), 159-208 [10.1007/S00440-022-01159-4].

Entropy-Transport distances between unbalanced metric measure spaces

De Ponti, N;
2022

Abstract

Inspired by the recent theory of Entropy-Transport problems and by the 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” 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.
Articolo in rivista - Articolo scientifico
metric measure spaces; unbalanced optimal transport; distances
English
10-ago-2022
2022
184
1-2
159
208
none
De Ponti, N., Mondino, A. (2022). Entropy-Transport distances between unbalanced metric measure spaces. PROBABILITY THEORY AND RELATED FIELDS, 184(1-2), 159-208 [10.1007/S00440-022-01159-4].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/462758
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