Wasserstein space over the Wiener space

• 成果类型: Article
• 署名作者: Fang, Shizan; Shao, Jinghai; Sturm, Karl-Theodor
• 署名单位: University of Bonn; Universite Bourgogne Europe; Beijing Normal University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-009-0199-5
• 发表日期: 2010
• 页码: 535-565
• 关键词: monge-ampere equation; transportation; geometry

摘要

The goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the Wiener space, based on and extending fundamental results of Feyel-stunel. Carrying out the program of Ambrosio-Gigli-Savar,, we present a complete characterization of the derivative processes for certain class of absolutely continuous curves. We prove existence of the gradient flow curves for the relative entropy w.r.t. the Wiener measure and identify these gradient flow curves with solutions of the Ornstein-Uhlenbeck evolution equation.