Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures

• 成果类型: Article
• 署名作者: Liero, Matthias; Mielke, Alexander; Savare, Giuseppe
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; University of Pavia
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0759-8
• 发表日期: 2018
• 页码: 969-1117
• 关键词: metric-measure-spaces; curvature-dimension condition; RICCI CURVATURE; INEQUALITY; EQUATIONS; geometry

摘要

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.