Benamou-Brenier and duality formulas for the entropic cost on RCD* (K, N) spaces

• 成果类型: Article
• 署名作者: Gigli, Nicola; Tamanini, Luca
• 署名单位: International School for Advanced Studies (SISSA); University of Bonn
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00909-1
• 发表日期: 2020
• 页码: 1-34
• 关键词: transport

摘要

In this paper we prove that, within the framework of RCD* (K, N) spaces with N < infinity, the entropic cost (i.e. the minimal value of the Schrodinger problem) admits: A threefold dynamical variational representation, in the spirit of the Benamou-Brenier formula for the Wasserstein distance; A Hamilton-Jacobi-Bellman dual representation, in line with Bobkov-Gentil-Ledoux and Otto-Villani results on the duality between Hamilton-Jacobi and continuity equation for optimal transport; A Kantorovich-type duality formula, where the Hopf-Lax semigroup is replaced by a suitable 'entropic' counterpart. We thus provide a complete and unifying picture of the equivalent variational representations of the Schrodinger problem as well as a perfect parallelism with the analogous formulas for the Wasserstein distance. Riemannian manifolds with Ricci curvature bounded from below are a relevant class of RCD* (K, N) spaces and our results are new even in this setting.