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July 2012 A spatial version of the Itô–Stratonovich correction
Martin Hairer, Jan Maas
Ann. Probab. 40(4): 1675-1714 (July 2012). DOI: 10.1214/11-AOP662


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.


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Martin Hairer. Jan Maas. "A spatial version of the Itô–Stratonovich correction." Ann. Probab. 40 (4) 1675 - 1714, July 2012.


Published: July 2012
First available in Project Euclid: 4 July 2012

zbMATH: 1262.60060
MathSciNet: MR2978135
Digital Object Identifier: 10.1214/11-AOP662

Primary: 60H15
Secondary: 35K55 , 60H30 , 60H35

Keywords: Itô–Stratonovich correction , spatial discretizations , Stochastic Burgers equation , Wiener Chaos

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 4 • July 2012
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