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. The framework covers the Kardar–Parisi–Zhang equation , the $\Phi _{3}^{4}$ equation and the parabolic Anderson model, all of which requires renormalization in order for it to be well-defined.

Definition

A regularity structure $(A,T,G)$ consists of:

• a subset $A$ of $\mathbb {R} ^{d}$ that's bounded from below and that's without accumulation points;
• the model space: a graded vector space $T=\oplus _{\alpha \in A}T_{\alpha }$, where each $T_{\alpha }$ is a Banach space;
• the structure group: a group $G$ of continuous operators $\Gamma$ such that, for each $\alpha \in A$ and each $\tau \in T_{\alpha }$, we have $(\Gamma -1)\tau \in \oplus _{\beta <\alpha }T_{\beta }$.