Martin Hairer's theory of regularity structures provides a framework for studying a large class of subcritical parabolic stochastic partial differential equations arising from quantum field theory.[1] The framework covers the Kardar–Parisi–Zhang equation, the equation and the parabolic Anderson model, all of which require renormalization in order to have a well-defined notion of solution.

Hairer won the 2021 Breakthrough Prize in mathematics for introducing regularity structures.[2]

Definition

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A regularity structure is a triple   consisting of:

  • a subset   (index set) of   that is bounded from below and has no accumulation points;
  • the model space: a graded vector space  , where each   is a Banach space; and
  • the structure group: a group   of continuous linear operators   such that, for each   and each  , we have  .

A further key notion in the theory of regularity structures is that of a model for a regularity structure, which is a concrete way of associating to any   and   a "Taylor polynomial" based at   and represented by  , subject to some consistency requirements. More precisely, a model for   on  , with   consists of two maps

 ,
 .

Thus,   assigns to each point   a linear map  , which is a linear map from   into the space of distributions on  ;   assigns to any two points   and   a bounded operator  , which has the role of converting an expansion based at   into one based at  . These maps   and   are required to satisfy the algebraic conditions

 ,
 ,

and the analytic conditions that, given any  , any compact set  , and any  , there exists a constant   such that the bounds

 ,
 ,

hold uniformly for all  -times continuously differentiable test functions   with unit   norm, supported in the unit ball about the origin in  , for all points  , all  , and all   with  . Here   denotes the shifted and scaled version of   given by

 .

References

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  1. ^ Hairer, Martin (2014). "A theory of regularity structures". Inventiones Mathematicae. 198 (2): 269–504. arXiv:1303.5113. Bibcode:2014InMat.198..269H. doi:10.1007/s00222-014-0505-4. S2CID 119138901.
  2. ^ Sample, Ian (2020-09-10). "UK mathematician wins richest prize in academia". The Guardian. ISSN 0261-3077. Retrieved 2020-09-13.