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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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.

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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

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