Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Che, Rui¹  Huang, Wen¹  Li, Yao²  Tetali, Prasad³ 

[1] Univ Sci & Technol China, Chinese Acad Sci, Wu Wen Tsun Key Lab Math, Hefei 230026, Anhui, Peoples R China

[2] Univ Massachusetts, Dept Math & Stat, Amherst, MA 01003 USA

[3] Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA

关键词: Fokker-Planck equation;    Gibbs density;    Graph;    Talagrand inequality;   

DOI  :  10.1016/j.jde.2016.05.003

来源: Elsevier

【 摘 要 】

In 2012, Chow, Huang, Li and Zhou [7] proposed the Fokker-Planck equations for the free energy on a finite graph, in which they showed that the corresponding Fokker-Planck equation is a nonlinear ODE defined on a Riemannian manifold of probability distributions. Different choices for inner products result in different Fokker-Planck equations. The unique global equilibrium of each equation is a Gibbs distribution. In this paper we proved that the exponential rate of convergence towards the global equilibrium of these Fokker-Planck equations. The rate is measured by both the decay of the L-2 norm and that of the (relative) entropy. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [7]. The first one is a local inequality, while the second is a global inequality with respect to the lower bound metric from [7]. (C) 2016 Published by Elsevier Inc.

【 授权许可】

Free