ENTROPIC MEASURE AND WASSERSTEIN DIFFUSION

• 成果类型: Article
• 署名作者: von Renesse, Max-K.; Sturm, Karl-Theodor
• 署名单位: Technical University of Berlin; University of Bonn
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP430
• 发表日期: 2009
• 页码: 1114-1191
• 关键词: quasi-invariance; diffeomorphism group; brownian-motion; fleming-viot; geometry; INEQUALITY; EQUATIONS; SPACES

摘要

We construct a new random probability measure on the circle and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval, respectively. The associated integration by parts formula is used to construct two classes of diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin's Brownian motion on the homeomorphism group. The second one is a probability valued stochastic perturbation of the heat flow, whose intrinsic metric is the quadratic Wasserstein distance. It may be regarded as the canonical diffusion process on the Wasserstein space.