Author(s):

Hairer, Martin M; Maas, Jan

Article Title: 
A spatial version of the ItôStratonovich 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 crossvariation (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: 
Itôstratonovich correction; Spatial discretizations; Stochastic burgers equation; Wiener chaos

Journal Title:

Annals of Probability

Volume: 
40

Issue 
4

ISSN:

00911798

Publisher:

Institute of Mathematical Statistics

Date Published:

20120701

Start Page: 
1675

End Page:

1714

Sponsor: 
Supported by Rubicon Grant 680500901 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: 
10.1214/11AOP662

Open access: 
yes (repository) 