Existence and stability for Fokker-Planck equations with log-concave reference measure

• 成果类型: Article
• 署名作者: Ambrosio, Luigi; Savare, Giuseppe; Zambotti, Lorenzo
• 署名单位: Universite Paris Cite; Sorbonne Universite; Scuola Normale Superiore di Pisa; University of Pavia
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0177-3
• 发表日期: 2009
• 页码: 517-564
• 关键词: phi interface model; convex measures; fluctuations; reflection; spdes; integration; motion; SPACES; parts; wall

摘要

We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker-Planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures, endowed with the Wasserstein distance.