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 $$\Gamma_{x y}$$, 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
 * $$\Gamma_{x y} \Gamma_{y z} = \Gamma_{x z}$$,
 * $$\Pi_{x} \Gamma_{x y} = \Pi_{y}$$,

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
 * $$| ( \Pi_{x} \tau ) \varphi_{x}^{\lambda} | \leq C \lambda^{|\tau|} \| \tau \|_{T_{\alpha}}$$,
 * $$\| \Gamma_{x y} \tau \|_{T_{\beta}} \leq C | x - y |^{\alpha - \beta} \| \tau \|_{T_{\alpha}}$$,

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