STRONG UNIQUENESS FOR STOCHASTIC EVOLUTION EQUATIONS IN HILBERT SPACES PERTURBED BY A BOUNDED MEASURABLE DRIFT

• 成果类型: Article
• 署名作者: Da Prato, G.; Flandoli, F.; Priola, E.; Roeckner, M.
• 署名单位: Scuola Normale Superiore di Pisa; University of Pisa; University of Turin; University of Bielefeld
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP763
• 发表日期: 2013
• 页码: 3306-3344
• 关键词: ornstein-uhlenbeck semigroups; pathwise uniqueness; sdes; EXISTENCE

摘要

We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on R-d to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions.