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Rough stochastic PDEs

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Hairer, Martin (2011) Rough stochastic PDEs. Communications on Pure and Applied Mathematics, Vol.64 (No.11). pp. 1547-1585. doi:10.1002/cpa.20383 ISSN 0010-3640.

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Official URL: http://dx.doi.org/10.1002/cpa.20383

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Abstract

In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too-high spatial roughness for classical analytical methods to apply. In fact, the class of SPDEs that we consider is genuinely ill-posed in the sense that different approximations to the nonlinearity may converge to different limits. Using rough path theory, a pathwise notion of solution to these SPDEs is formulated, and we show that this yields a well-posed problem that is stable under a large class of perturbations, including the approximation of the rough-driving noise by a mollified version and the addition of hyperviscosity.

We also show that under certain structural assumptions on the coefficients, the SPDEs under consideration generate a reversible Markov semigroup with respect to a diffusion measure that can be given explicitly.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Q Science > QC Physics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Differential equations, Partial, Geometric quantization, Stochastic partial differential equations
Journal or Publication Title: Communications on Pure and Applied Mathematics
Publisher: John Wiley & Sons
ISSN: 0010-3640
Official Date: November 2011
Dates:
DateEvent
November 2011Published
Volume: Vol.64
Number: No.11
Page Range: pp. 1547-1585
DOI: 10.1002/cpa.20383
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Funder: Engineering and Physical Sciences Research Council (EPSRC), Royal Society (Great Britain), Leverhulme Trust (LT)
Grant number: EP/E002269/1 (EPSRC), EP/D071593/1 (EPSRC)

Data sourced from Thomson Reuters' Web of Knowledge

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