Rough stochastic PDEs
Hairer, Martin. (2011) Rough stochastic PDEs. Communications on Pure and Applied Mathematics, Vol.64 (No.11). pp. 1547-1585. ISSN 0010-3640Full text not available from this repository.
Official URL: http://dx.doi.org/10.1002/cpa.20383
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|
|Official Date:||November 2011|
|Page Range:||pp. 1547-1585|
|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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