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A theory of regularity structures

Originality
+ 5 - 0
Accuracy
+ 4 - 0
Score
10.87
3664 views
Referee this paper: arXiv:1303.5113 by M. Hairer

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

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This 2013 paper by M. Hairer introduces a new notion of "regularity structure" that provides an algebraic framework allowing to describe functions and / or distributions via a kind of "jet" or local Taylor expansion around each point.

''The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. 
We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a "renormalisation group" which is defined canonically in terms of the regularity structure associated to the given class of PDEs. 
As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the $\Phi^4_3$ Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of 3-dimensional ferromagnets near their critical temperature.''

summarized by Arnold Neumaier
paper authored Mar 20, 2013 to Reviews I by  (no author on PO assigned yet) 
  • [ revision history ]
    retagged Aug 31, 2014 by dimension10

    Yesterday (August 12, 2014), Martin Hairer received the Fields medal for this work; see 
    http://www.mathunion.org/fileadmin/IMU/Prizes/2014/news_release_hairer.pdf

    The fields medal is (together with the Abel prize) the highest honor a mathematician can obtain.

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