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Approximating rough stochastic PDEs
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Hairer, Martin, Maas, Jan and Weber, Hendrik (2014) Approximating rough stochastic PDEs. Communications on Pure and Applied Mathematics, Volume 67 (Number 5). pp. 776-870. doi:10.1002/cpa.21495 ISSN 0010-3640.
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Official URL: http://dx.doi.org/10.1002/cpa.21495
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.
Item Type: | Journal Article | ||||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||
Journal or Publication Title: | Communications on Pure and Applied Mathematics | ||||||||||
Publisher: | John Wiley & Sons | ||||||||||
ISSN: | 0010-3640 | ||||||||||
Official Date: | May 2014 | ||||||||||
Dates: |
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Volume: | Volume 67 | ||||||||||
Number: | Number 5 | ||||||||||
Page Range: | pp. 776-870 | ||||||||||
DOI: | 10.1002/cpa.21495 | ||||||||||
Status: | Peer Reviewed | ||||||||||
Publication Status: | Published | ||||||||||
Access rights to Published version: | Restricted or Subscription Access |
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