A spatial version of the Itô-Stratonovich correction Journal Article


Author(s): Hairer, Martin M; Maas, Jan
Article Title: A spatial version of the Itô-Stratonovich correction
Affiliation
Abstract: 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 Itô–Stratonovich correction term that arises when one considers different temporal discretizations of stochastic ODEs.
Keywords: Itô-stratonovich correction; Spatial discretizations; Stochastic burgers equation; Wiener chaos
Journal Title: Annals of Probability
Volume: 40
Issue 4
ISSN: 0091-1798
Publisher: Institute of Mathematical Statistics  
Date Published: 2012-07-01
Start Page: 1675
End Page: 1714
Sponsor: Supported by Rubicon Grant 680-50-0901 of the Netherlands Organisation for Scientific Research (NWO). Supported by the EPSRC Grants EP/E002269/1 and EP/D071593/1, a Wolfson Research Merit Award of the Royal Society and a Philip Leverhulme prize of the Lev
URL:
DOI: 10.1214/11-AOP662
Open access: yes (repository)
IST Austria Authors
  1. Jan Maas
    28 Maas
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