A SPATIAL VERSION OF THE ITO-STRATONOVICH CORRECTION

• 成果类型: Article
• 署名作者: Hairer, Martin; Maas, Jan
• 署名单位: University of Warwick; University of Bonn
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP662
• 发表日期: 2012
• 页码: 1675-1714
• 关键词: stochastic burgers; EQUATIONS

摘要

We consider a class of stochastic PDEs of Burgers type in spatial dimension 1, driven by space-time white noise. Even though it is well known that these equations are well posed, it turns out that if one performs a spatial discretization of the nonlinearity in the wrong way, then the sequence of approximate equations does converge to a limit, but this limit exhibits an additional correction term. This correction term is proportional to the local quadratic cross-variation (in space) of the gradient of the conserved quantity with the solution itself. This can be understood as a consequence of the fact that for any fixed time, the law of the solution is locally equivalent to Wiener measure, where space plays the role of time. In this sense, the correction term is similar to the usual Ito Stratonovich correction term that arises when one considers different temporal discretizations of stochastic ODEs.