Displacement convexity of entropy and related inequalities on graphs

• 成果类型: Article
• 署名作者: Gozlan, Nathael; Roberto, Cyril; Samson, Paul-Marie; Tetali, Prasad
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel; Universite Paris Nanterre; University System of Georgia; Georgia Institute of Technology; University System of Georgia; Georgia Institute of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0523-y
• 发表日期: 2014
• 页码: 47-94
• 关键词: logarithmic sobolev inequalities; metric-measure-spaces; RICCI CURVATURE; markov-chains; PROBABILITY-MEASURES; transportation; INFORMATION; bernoulli; brascamp; geometry

摘要

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Pr,kopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal-by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.