A theory of regularity structures
Abstract
We introduce 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 purposebuilt 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 longstanding 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 3dimensional ferromagnets near their critical temperature.
 Publication:

Inventiones Mathematicae
 Pub Date:
 November 2014
 DOI:
 10.1007/s0022201405054
 arXiv:
 arXiv:1303.5113
 Bibcode:
 2014InMat.198..269H
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 Mathematics  Probability;
 60H15;
 81S20;
 82C28
 EPrint:
 180 pages