Entropic curvature on graphs along Schrodinger bridges at zero temperature

• 成果类型: Article
• 署名作者: Samson, Paul-Marie
• 署名单位: Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01167-4
• 发表日期: 2022
• 页码: 859-937
• 关键词: metric-measure-spaces; RICCI CURVATURE; discrete; transport; inequalities; convexity; geometry; bounds; cost

摘要

Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. Leonard by replacing W-2-Wasserstein geodesics by Schrodinger bridges in the definition of entropic curvature (Leonard in Discrete Contin Dyn Syst A 34(4):1533-1574, 2014; Ann Probab 44(3):1864-1915, 2016; in: Gigli N (ed) Measure theory in non-smooth spaces. Sciendo Migration,Warsaw, pp 194-242, 2017). As a remarkable fact, as a temperature parameter goes to zero, these Schrodinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graph spaces: the lattice Z endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli-Laplace model. Our general results also apply to a large class of graphs which are not specifically studied in this paper. As opposed to Erbar-Maas results on graphs (Erbar and Maas in Arch Ration Mech Anal 206(3):997-1038, 2012; Discrete Contin Dyn Syst A 34(4):1355-1374, 2014; Maas in J Funct Anal 261(8):2250-2292, 2011), entropic curvature results of this paper imply new Prekopa-Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube {0, 1} and for the Bernoulli Laplace model, a new W-2 - W-1 transport-entropy inequality is reached, that can not be derived by usual induction arguments over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see Gozlan et al. in J Funct Anal 273(11):3327-3405, 2017) associated to the so-called convex-hull method by Talagrand (Publ Math l'Inst Hautes Etudes Sci 81(1):73-205, 1995).