A spatial version of the Itō-Stratonovich correction. (English) Zbl 1262.60060

Hairer, Martin; Maas, Jan

Ann. Probab. 40, No. 4, 1675-1714 (2012).

We consider a class of stochastic PDEs of Burgers type in spatial dimension one, 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.
The correction term is proportional to the local quadratic cross-variation 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 palys the role of time. In this sense, the correction term is similar to the usual Itō-Stratonovich correction term that arises when one considers differential temporal discretization of stochastic ODEs.

Reviewer: Kai Liu (Liverpool)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35K55	Nonlinear parabolic equations
60H30	Applications of stochastic analysis (to PDEs, etc.)

Keywords:

stochastic Burgers equations

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