# A theory of regularity structures

@article{Hairer2014ATO, title={A theory of regularity structures}, author={Martin Hairer}, journal={Inventiones mathematicae}, year={2014}, volume={198}, pages={269-504} }

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…

## 713 Citations

An analytic BPHZ theorem for regularity structures

- Mathematics
- 2016

We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated…

Algebraic renormalisation of regularity structures

- MathematicsInventiones mathematicae
- 2018

We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a…

Paracontrolled calculus and regularity structures I

- MathematicsJournal of the Mathematical Society of Japan
- 2018

We start in this work the study of the relation between the theory of regularity structures and paracontrolled calculus. We give a paracontrolled representation of the reconstruction operator and…

Stochastic PDEs, Regularity Structures, and Interacting Particle Systems

- Mathematics
- 2015

These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of…

A Solution Theory for Quasilinear Singular SPDEs

- MathematicsCommunications on Pure and Applied Mathematics
- 2019

We give a construction allowing us to build local renormalized solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalizing the recent…

Ito formulae for the stochastic heat equation via the theories of rough paths and regularity structures

- Mathematics
- 2019

In this thesis, we develop a general theory to prove the existence of several Ito formulae on the one-dimensional stochastic heat equation driven by additive space-time white noise. That is denoting…

Modelled distributions of Triebel–Lizorkin type

- MathematicsStudia Mathematica
- 2020

In order to provide a local description of a regular function in a small neighbourhood of a point $x$, it is sufficient by Taylor's theorem to know the value of the function as well as all of its…

A diagram-free approach to the stochastic estimates in regularity structures

- Computer Science
- 2021

This paper construct and stochastically estimate the renormalized model postulated in [32], avoiding the use of Feynman diagrams but still in a fully automated way and capturing the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modelled distribution.

Approximations of dispersive PDEs in the presence of low-regularity randomness

- Mathematics, Computer ScienceArXiv
- 2022

A novel combinatorial structure called paired decorated forests which are two decorated trees whose decorations on the leaves come in pair are introduced which allows for low regularity approximations to the expectation E ( | u k ( τ, v η ) | 2 ) , where u k denotes the k -th Fourier coeﬃcient of the solution u of the dispersive equation and v x the associated random initial data.

Online First version Modelled distributions of Triebel – Lizorkin type by

- Mathematics
- 2019

In order to provide a local description of a regular function in a small neighbourhood of a point x, it is sufficient by Taylor’s theorem to know the value of the function as well as all of its…

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