Regularity structure explained

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

\Phi ^{4} _{3} 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

l{T} = (A,T,G) consisting of:

• a subset

A (index set) of R that is bounded from below and has no accumulation points;T = ⊕ _{\alpha} T _{\alpha} , where each T _{\alpha} is a Banach space; andG 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 ⊕ _{\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 R ^{d} a "Taylor polynomial" based at x _{0} and represented by \tau , subject to some consistency requirements.More precisely, a model for l{T} = (A,T,G) on R ^{d} , with d \geq 1 consists of two maps\Pi \colon R ^{d} \to Lin (T; l{S}'( R ^{d} )) ,\Gamma \colon R ^{d} x 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 R ^{d} ; \Gamma assigns to any two points x and y a bounded operator \Gamma _{x} , 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} \Gamma _{y} = \Gamma _{x} ,\Pi _{x} \Gamma _{x} = \Pi _{y} ,and the analytic conditions that, given any r > | inf A | , any compact set K \subset R ^{d} , and any \gamma > 0 , there exists a constant C > 0 such that the bounds| ( \Pi _{x} \tau ) \varphi ^{λ} _{x} | \leq C λ ^{|\tau|} \| \tau \| _{T _{\alpha}} ,\| \Gamma _{x} \tau \| _{T _{\beta}} \leq C | x - y | ^{\alpha} \| \tau \| _{T _{\alpha}} ,hold uniformly for all r -times continuously differentiable test functions \varphi \colon R ^{d} \to R with unit l{C} ^{r} norm, supported in the unit ball about the origin in R ^{d} , for all points x, y \in K , all 0 < λ \leq 1 , and all \tau \in T _{\alpha} with \beta < \alpha \leq \gamma . Here \varphi ^{λ} _{x} \colon R ^{d} \to R denotes the shifted and scaled version of \varphi given by\varphi ^{λ} _{x} (y) = λ ^{-d} \varphi \left( (y - x)/(λ) \right) .

Notes and References

1. Hairer. Martin. A theory of regularity structures. Inventiones Mathematicae. 2014. 10.1007/s00222-014-0505-4. 198. 2. 269–504. 1303.5113. 2014InMat.198..269H. 119138901.
2. News: Sample. Ian . 2020-09-10. UK mathematician wins richest prize in academia. en-GB. The Guardian. 2020-09-13. 0261-3077.