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Hairer, Martin. (2011) Rough stochastic PDEs. Communications on Pure and Applied Mathematics, Vol.64 (No.11). pp. 1547-1585. 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 |
| Divisions: | Faculty of 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 |
| Date: | November 2011 |
| Volume: | Vol.64 |
| Number: | No.11 |
| Page Range: | pp. 1547-1585 |
| Identification Number: | 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) |
| URI: | http://wrap.warwick.ac.uk/id/eprint/38926 |
Data sourced from Thomson Reuters' Web of Knowledge
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