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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
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 |
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| 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: |
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| 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) |
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