Hairer, Martin M; Maas, JanIST Austria
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.
Annals of Probability
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
1675 - 1714
Hairer M, Maas J. A spatial version of the Itô-Stratonovich correction. Annals of Probability. 2012;40(4):1675-1714. doi:10.1214/11-AOP662
Hairer, M., & Maas, J. (2012). A spatial version of the Itô-Stratonovich correction. Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/11-AOP662
Hairer, Martin, and Jan Maas. “A Spatial Version of the Itô-Stratonovich Correction.” Annals of Probability. Institute of Mathematical Statistics, 2012. https://doi.org/10.1214/11-AOP662.
M. Hairer and J. Maas, “A spatial version of the Itô-Stratonovich correction,” Annals of Probability, vol. 40, no. 4. Institute of Mathematical Statistics, pp. 1675–1714, 2012.
Hairer M, Maas J. 2012. A spatial version of the Itô-Stratonovich correction. Annals of Probability. 40(4), 1675–1714.
Hairer, Martin, and Jan Maas. “A Spatial Version of the Itô-Stratonovich Correction.” Annals of Probability, vol. 40, no. 4, Institute of Mathematical Statistics, 2012, pp. 1675–714, doi:10.1214/11-AOP662.