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. The framework covers the Kardar–Parisi–Zhang equation, the $\Phi _{3}^{4}$ 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.

Definition

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

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

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

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

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

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

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

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

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

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

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

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

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