Physics:Regularity structure

Short description: Framework for studying stochastic partial differential equations

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 ${\mathrm{\Phi }}_3^4$ 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 ${𝒯}=(A,T,G)$ 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 $\mathrm{\Gamma }:T\to T$ such that, for each $\alpha \in A$ and each $\tau \in T_{\alpha }$, we have $(\mathrm{\Gamma }-1)\tau \in {\oplus }_{\beta <\alpha }{T}_{\beta }$.

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 ${𝒯}=(A,T,G)$ on ${\mathbb{ℝ}}^d$, with $d\ge 1$ consists of two maps

$\mathrm{\Pi }:{\mathbb{ℝ}}^d\to \mathrm{L}\mathrm{i}\mathrm{n}(T;{{𝒮}}^{\prime }({\mathbb{ℝ}}^d))$,

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

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

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

${\mathrm{\Pi }}_x{\mathrm{\Gamma }}_{xy}={\mathrm{\Pi }}_y$,

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

$|({\mathrm{\Pi }}_x\tau ){\phi }_x^{\lambda }|\le C{\lambda }^{|\tau |}\Vert \tau {\Vert }_{T_{\alpha }}$,

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

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

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