Inventiones mathematicae

, Volume 198, Issue 2, pp 269–504 | Cite as

A theory of regularity structures

  • M. HairerEmail author


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 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. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. 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.

Mathematics Subject Classification (2000)

60H15 81S20 82C28 



I am very grateful to M. Gubinelli and to H. Weber for our numerous discussions on quantum field theory, renormalisation, rough paths, paraproducts, Hopf algebras, etc. These discussions were of enormous help in clarifying the concepts presented in this article. Many other people provided valuable input that helped shaping the theory. In particular, I would like to mention A. Debussche, B. Driver, P. Friz, J. Jones, D. Kelly, X.-M. Li, M. Lewin, T. Lyons, J. Maas, K. Matetski, J.-C. Mourrat, N. Pillai, D. Simon, T. Souganidis, J. Unterberger, and L. Zambotti. Special thanks are due to L. Zambotti for pointing out a mistake in an earlier version of the definition of the class of regularity structures considered in Sect. 8. Financial support was kindly provided by the Royal Society through a Wolfson Research Merit Award and by the Leverhulme Trust through a Philip Leverhulme Prize.


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Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of WarwickCoventryU.K

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