Entropy-Transport distances between unbalanced metric measure spaces

• 成果类型: Article
• 署名作者: De Ponti, Nicolo; Mondino, Andrea
• 署名单位: International School for Advanced Studies (SISSA); University of Oxford
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01159-4
• 发表日期: 2022
• 页码: 159-208
• 关键词: hellinger-kantorovich distance; RICCI CURVATURE

摘要

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.