Approximating Rough Stochastic PDEs Journal Article

Author(s): Hairer, Martin M; Maas, Jan; Weber, Hendrik
Article Title: Approximating Rough Stochastic PDEs
Abstract: We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô-Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.
Journal Title: Communications on Pure and Applied Mathematics
Volume: 67
Issue 5
ISSN: 1097-0312
Publisher: Wiley-Blackwell  
Date Published: 2014-05-01
Start Page: 776
End Page: 870
Sponsor: JM is supported by Rubicon grant 680-50-0901 of the Netherlands Organisation for Scientific Research (NWO). MH is supported by EPSRC grant EP/D071593/1 and by the Royal Society through a Wolfson Research Merit Award. Both MH and HW are supported by the Le
DOI: 10.1002/cpa.21495
Open access: yes (repository)