CONSERVATIVE STOCHASTIC TWO-DIMENSIONAL CAHN-HILLIARD EQUATION

• 成果类型: Article
• 署名作者: Rockner, Michael; Yang, Huanyu; Zhu, Rongchan
• 署名单位: University of Bielefeld; Free University of Berlin; Beijing Institute of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1620
• 发表日期: 2021
• 页码: 1336-1375
• 关键词: spectral gap; QUANTIZATION; MODEL

摘要

We consider the stochastic two-dimensional Cahn-Hilliard equation which is driven by the derivative in space of a space-time white noise. We use two different approaches to study this equation. First we prove that there exists a unique solution Y to the shifted equation (1.4). Then X := Y + Z is the unique solution to the stochastic Cahn-Hilliard equation, where Z is the corresponding O-U process. Moreover, we use the Dirichlet form approach in (Probab. Theory Related Fields 89 (1991) 347-386) to construct a probabilistically weak solution to the original equation (1.1) below. By clarifying the precise relation between the two solutions, we also get the restricted Markov uniqueness of the generator and the uniqueness of the martingale solutions to the equation (1.1). Furthermore, we also obtain exponential ergodicity of the solutions.