regularity structures

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 $\mathcal = (A,T,G)$ consisting of:
* a subset $A$ (index set) of $\mathbb$ that is bounded from below and has no accumulation points;
* the model space: a graded vector space $T = \oplus_ T_$, where each $T_$ 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 $\tau \in T_$, we have $(\Gamma-1)\tau \in \oplus_ T_$.

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 $x_ \in \mathbb^$ a "Taylor polynomial" based at $x_$ and represented by $\tau$, subject to some consistency requirements.
More precisely, a model for $\mathcal = (A,T,G)$ on $\mathbb^$, with $d \geq 1$ consists of two maps
:$\Pi \colon \mathbb^ \to \mathrm(T; \mathcal'(\mathbb^))$,
:$\Gamma \colon \mathbb^ \times \mathbb^ \to G$.
Thus, $\Pi$ assigns to each point $x$ a linear map $\Pi_$, which is a linear map from $T$ into the space of distributions on $\mathbb^$; $\Gamma$ assigns to any two points $x$ and $y$ a bounded operator $\Gamma_$, 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
:$\Gamma_ \Gamma_ = \Gamma_$,
:$\Pi_ \Gamma_ = \Pi_$,
and the analytic conditions that, given any $r > , \inf A ,$, any compact set $K \subset \mathbb^$, and any $\gamma > 0$, there exists a constant $C > 0$ such that the bounds
:$, ( \Pi_ \tau ) \varphi_^ , \leq C \lambda^ \, \tau \, _$,
:$\, \Gamma_ \tau \, _ \leq C , x - y , ^ \, \tau \, _$,
hold uniformly for all $r$-times continuously differentiable test functions $\varphi \colon \mathbb^ \to \mathbb$ with unit $\mathcal^$ norm, supported in the unit ball about the origin in $\mathbb^$, for all points $x, y \in K$, all $0 < \lambda \leq 1$, and all $\tau \in T_$ with $\beta < \alpha \leq \gamma$. Here $\varphi_^ \colon \mathbb^ \to \mathbb$ denotes the shifted and scaled version of $\varphi$ given by
:$\varphi_^ (y) = \lambda^ \varphi \left( \frac \right)$.

References

Stochastic differential equations
Quantum field theory
Statistical mechanics

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