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 [math]\displaystyle{ \Phi_3^4 }[/math] 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 [math]\displaystyle{ \mathcal{T} = (A,T,G) }[/math] consisting of:

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

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

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

[math]\displaystyle{ \Gamma \colon \mathbb{R}^{d} \times \mathbb{R}^{d} \to G }[/math].

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

[math]\displaystyle{ \Gamma_{x y} \Gamma_{y z} = \Gamma_{x z} }[/math],

[math]\displaystyle{ \Pi_{x} \Gamma_{x y} = \Pi_{y} }[/math],

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

[math]\displaystyle{ | ( \Pi_{x} \tau ) \varphi_{x}^{\lambda} | \leq C \lambda^{|\tau|} \| \tau \|_{T_{\alpha}} }[/math],

[math]\displaystyle{ \| \Gamma_{x y} \tau \|_{T_{\beta}} \leq C | x - y |^{\alpha - \beta} \| \tau \|_{T_{\alpha}} }[/math],

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

[math]\displaystyle{ \varphi_{x}^{\lambda} (y) = \lambda^{-d} \varphi \left( \frac{y - x}{\lambda} \right) }[/math].