Solving the KPZ equation

Publication available at: https://arxiv.org/pdf/1109.6811
Title: Solving the KPZ equation
Author(s): Hairer, M
Item Type: Journal Article
Abstract: We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a “universal” measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class. As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the Fokker-Planck equation associated to a particle diffusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.
Publication Date: 1-Sep-2013
Date of Acceptance: 28-Jun-2012
URI: http://hdl.handle.net/10044/1/55729
DOI: https://dx.doi.org/10.4007/annals.2013.178.2.4
ISSN: 1939-8980
Publisher: Princeton University, Department of Mathematics
Start Page: 559
End Page: 664
Journal / Book Title: Annals of Mathematics
Volume: 178
Issue: 2
Copyright Statement: © 2013 Department of Mathematics, Princeton University.
Copyright Statement: © 2013 Department of Mathematics, Princeton University.
Keywords: Science & Technology
Physical Sciences
Mathematics
PARTIAL-DIFFERENTIAL-EQUATIONS
ROUGH PATHS
INITIAL CONDITION
DRIVEN
HOMOGENIZATION
BURGERS
NOISE
DISTRIBUTIONS
COEFFICIENTS
SIGNALS
Science & Technology
Physical Sciences
Mathematics
PARTIAL-DIFFERENTIAL-EQUATIONS
ROUGH PATHS
INITIAL CONDITION
DRIVEN
HOMOGENIZATION
BURGERS
NOISE
DISTRIBUTIONS
COEFFICIENTS
SIGNALS
0101 Pure Mathematics
General Mathematics
Publication Status: Published
Open Access location: https://arxiv.org/pdf/1109.6811
Appears in Collections:Pure Mathematics
Mathematics



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