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.^[1] The framework covers the Kardar–Parisi–Zhang equation, the ${\displaystyle \Phi _{3}^{4}}$ 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.^[2]

Definition

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

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

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

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

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

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

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

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

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

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

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

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

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

References

1. Hairer, Martin (2014). "A theory of regularity structures". Inventiones Mathematicae. 198 (2): 269–504. arXiv:1303.5113. Bibcode:2014InMat.198..269H. doi:10.1007/s00222-014-0505-4. S2CID 119138901.
2. Sample, Ian (2020-09-10). "UK mathematician wins richest prize in academia". The Guardian. ISSN 0261-3077. Retrieved 2020-09-13.