Regularity structure

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 ${\displaystyle \Phi _{3}^{4}}$ 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

A regularity structure is a triple ${\displaystyle {\mathcal {T}}=(A,T,G)}$ consisting of:

• a subset ${\displaystyle A}$ (index set) of ${\displaystyle \mathbb {R} }$ that is bounded from below and has no accumulation points;
• the model space: a graded vector space ${\displaystyle T=\oplus _{\alpha \in A}T_{\alpha }}$, where each ${\displaystyle T_{\alpha }}$ is a Banach space; and
• the structure group: a group ${\displaystyle G}$ of continuous linear operators ${\displaystyle \Gamma \colon T\to T}$ such that, for each ${\displaystyle \alpha \in A}$ and each ${\displaystyle \tau \in T_{\alpha }}$, we have ${\displaystyle (\Gamma -1)\tau \in \oplus _{\beta <\alpha }T_{\beta }}$.

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 ${\displaystyle \tau \in T}$ and ${\displaystyle x_{0}\in \mathbb {R} ^{d}}$ a "Taylor polynomial" based at ${\displaystyle x_{0}}$ and represented by ${\displaystyle \tau }$, subject to some consistency requirements. More precisely, a model for ${\displaystyle {\mathcal {T}}=(A,T,G)}$ on ${\displaystyle \mathbb {R} ^{d}}$, with ${\displaystyle d\geq 1}$ consists of two maps

${\displaystyle \Pi \colon \mathbb {R} ^{d}\to \mathrm {Lin} (T;{\mathcal {S}}'(\mathbb {R} ^{d}))}$,

${\displaystyle \Gamma \colon \mathbb {R} ^{d}\times \mathbb {R} ^{d}\to G}$.

Thus, ${\displaystyle \Pi }$ assigns to each point ${\displaystyle x}$ a linear map ${\displaystyle \Pi _{x}}$, which is a linear map from ${\displaystyle T}$ into the space of distributions on ${\displaystyle \mathbb {R} ^{d}}$; ${\displaystyle \Gamma }$ assigns to any two points ${\displaystyle x}$ and ${\displaystyle y}$ a bounded operator ${\displaystyle \Gamma _{xy}}$, which has the role of converting an expansion based at ${\displaystyle y}$ into one based at ${\displaystyle x}$. These maps ${\displaystyle \Pi }$ and ${\displaystyle \Gamma }$ are required to satisfy the algebraic conditions

${\displaystyle \Gamma _{xy}\Gamma _{yz}=\Gamma _{xz}}$,

${\displaystyle \Pi _{x}\Gamma _{xy}=\Pi _{y}}$,

and the analytic conditions that, given any ${\displaystyle r>|\inf A|}$, any compact set ${\displaystyle K\subset \mathbb {R} ^{d}}$, and any ${\displaystyle \gamma >0}$, there exists a constant ${\displaystyle C>0}$ such that the bounds

${\displaystyle |(\Pi _{x}\tau )\varphi _{x}^{\lambda }|\leq C\lambda ^{|\tau |}\|\tau \|_{T_{\alpha }}}$,

${\displaystyle \|\Gamma _{xy}\tau \|_{T_{\beta }}\leq C|x-y|^{\alpha -\beta }\|\tau \|_{T_{\alpha }}}$,

hold uniformly for all ${\displaystyle r}$-times continuously differentiable test functions ${\displaystyle \varphi \colon \mathbb {R} ^{d}\to \mathbb {R} }$ with unit ${\displaystyle {\mathcal {C}}^{r}}$ norm, supported in the unit ball about the origin in ${\displaystyle \mathbb {R} ^{d}}$, for all points ${\displaystyle x,y\in K}$, all ${\displaystyle 0<\lambda \leq 1}$, and all ${\displaystyle \tau \in T_{\alpha }}$ with ${\displaystyle \beta <\alpha \leq \gamma }$. Here ${\displaystyle \varphi _{x}^{\lambda }\colon \mathbb {R} ^{d}\to \mathbb {R} }$ denotes the shifted and scaled version of ${\displaystyle \varphi }$ given by

${\displaystyle \varphi _{x}^{\lambda }(y)=\lambda ^{-d}\varphi \left({\frac {y-x}{\lambda }}\right)}$.

References