THE STOCHASTIC REFLECTION PROBLEM ON AN INFINITE DIMENSIONAL CONVEX SET AND BV FUNCTIONS IN A GELFAND TRIPLE

• 成果类型: Article
• 署名作者: Roeckner, Michael; Zhu, Rong-Chan; Zhu, Xiang-Chan
• 署名单位: University of Bielefeld; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; Peking University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP661
• 发表日期: 2012
• 页码: 1759-1794
• 关键词: equations; THEOREM; spdes

摘要

In this paper, we introduce a definition of BV functions in a Gelfand triple which is an extension of the definition of BV functions in [Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur Rend. Lincei (9) Mat. Appl. 21 (2010) 405-414] by using Dirichlet form theory. By this definition, we can consider the stochastic reflection problem associated with a self-adjoint operator A and a cylindrical Wiener process on a convex set Gamma in a Hilbert space H. We prove the existence and uniqueness of a strong solution of this problem when Gamma is a regular convex set. The result is also extended to the nonsymmetric case. Finally, we extend our results to the case when Gamma = K-alpha, where K-alpha = {f is an element of L-2(0, 1)vertical bar f >= -alpha), alpha > 0.