Approximating Rough Stochastic PDEs

Martin Hairer, Jan Maas, Hendrik Weber

Research output: Contribution to journalArticle

18 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)776-870
Number of pages95
JournalCommunications on Pure and Applied Mathematics
Volume67
Issue number5
Early online date2 Dec 2013
DOIs
Publication statusPublished - 1 May 2014

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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