A theory of regularity structures

By Martin Hairer. 2013

1,123 citations recorded by the source corpus.

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 $$\Phi ^4_3$$Φ34 Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of $$3$$3-dimensional ferromagnets near their critical temperature.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/730812ae1de6e5acf7f39b5cc6ccf59f343d45ca

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

In Statistics and Probability

Citations in the corpus

HIGH ORDER PARACONTROLLED CALCULUS (2016, background): “[5] Y. Bruned and A. Chandra and I. Chevyrev and M. Hairer, Renormalising SPDEs in regularity struc- tures.”
Energy solutions of KPZ are unique (2015, methodology): “This difficulty persists also in other recent (but, in this case, mostly equivalent) methods of giving a meaning to the SBE: Hairer’s regularity structures [8] or Gubinelli, Imkeller and Perkowski’s paracontrolled distributions [6, 5].”
Spectral gap for the stochastic quantization equation on the 2-dimensional torus (2016, neutral): “low for a natural approximation of the objects deﬁned in (2.7). At this point we would like to mention that one might work directly with n 1 ;instead of introducing (2.7) (see for example [dPD03] and [Ha14]). This alternative approach has the advantage that the diagrams are stationary in time. Howev”
An analytic BPHZ theorem for regularity structures (2016, methodology): “ interfaces, INTRODUCTION 3 where F(’;) = jr’j2+, and the 4 dequation appearing from limits of dynamical Ising models, where F(’;) = ’3 + . An important restriction on the class of equations that [Hai14] is able to treat is local subcriticality – a precise deﬁnition of this condition is fairly techn”
The KPZ fixed point (2016, background): “s, and their description relies to a large degree on symmetry (often conformal invariance). In the case of ˚4 d , the main tool is perturbation theory. Even the recent theory of regularity structures [Hai14], which makes sense of the KPZ equation (1.2), does so by treating the non-linear term as a k”
A regularity structure for rough volatility (2017, methodology): “Example 3.6), then the renormalization 𝒞𝜀 = 𝒦𝜀(⋅, ⋅) that was applied in Theorem 3.14 and thus in Definition 3.24 is a constant, which is the familiar concept one encounters in the study of singular SPDE (Chandra & Hairer, 2016; Hairer, 2013, 2014).”
Renormalization Group and Stochastic PDEs (2016, methodology): “We prove almost sure local well-posedness for the mild (integral equation) version of (1) thereby recovering the results in [1].”
A Wong-Zakai theorem for stochastic PDEs (2014, methodology): “gence result given in Theorem 4.7 below. The statement of this stronger result does however ﬁrst require the introduction of the algebraic / analytical machinery of regularity structures developed in [Hai14b], so we refrain from stating it here. As a consequence of this stronger result, one immediat”
A priori estimates for rough PDEs with application to rough conservation laws (2016, background): “Hairer’s regularity structure theory [36] is a wide generalization of rough path which allows irregular objects parametrized by multidimensional indices.”
Heat semigroup and singular PDEs (2015, methodology): “Following the recent breakthrough of Hairer [25] and Gubinelli, Imkeller, Perkowski [22], there has been recently a tremendous activity in the study of parabolic singular Date: January 28, 2015.”
PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES (2012, background): “e solutions to the modiﬁed problem Luε= F(uε)ξε−F′(uε)F(uε)|Dxϑε|2 will converge as soon as ξε→ ξin Cα−2, without any requirements on the bilinear term ϑε◦ξε. 6 Relation with regularity structures In [Hai14] Hairer introduces a general framework that allows to describe distributions which locally be”
Multiplicative stochastic heat equations on the whole space (2015, methodology): “ation. This yields the convergence of the sequence of renormalised solutions uˆǫto a limit u. Let us now outline some major modiﬁcations that we bring to th e original theory of regularity structures [Hai14b]. First, since we want to start (E) from a Dirac mass, we need to choose an appropriate spac”
Restricted Markov uniqueness for the stochastic quantization of $P(\Phi)_2$ and its applications (2015, methodology): “see Theorem 4.7). We also want to mention that recently there has arisen a renewed interest in SPDEs related to such problems, particularly in connection with Hairer’s theory of regularity structures [Hai14] and related work by Imkeller, Gubinelli, Perkowski in [GIP13]. By using these theories one c”