The Annals of Probability

A spatial version of the Itô–Stratonovich correction

Martin Hairer and Jan Maas

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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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Ann. Probab. Volume 40, Number 4 (2012), 1675-1714.

First available in Project Euclid: 4 July 2012

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Mathematical Reviews number (MathSciNet)

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35K55: Nonlinear parabolic equations 60H30: Applications of stochastic analysis (to PDE, etc.) 60H35: Computational methods for stochastic equations [See also 65C30]

Itô–Stratonovich correction stochastic Burgers equation spatial discretizations Wiener chaos


Hairer, Martin; Maas, Jan. A spatial version of the Itô–Stratonovich correction. Ann. Probab. 40 (2012), no. 4, 1675--1714. doi:10.1214/11-AOP662.

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