Regularity structure

A regularity structure is an algebraic framework for local expansions that can include singular distributions as well as smooth functions. Its formal data are $(A,T,G)$: a locally finite set $A\subset \mathbb{R}$ containing zero and bounded below; a graded vector space $T={⨁}_{\alpha \in A}{T}_{\alpha }$ of Banach spaces, with $T_0$ the constants; and a group $G$ of linear maps fixing $1$ such thatThus changing an expansion's base point modifies a symbol only through lower-homogeneity terms. Martin Hairer introduced this framework to treat classes of singular stochastic partial differential equations.^[ref]

Ordinary polynomials provide a complete elementary example. In one dimension take $A=\{0,1,2,\dots \}$, $T_n$ spanned by $X^n$, and realize $X^n$ around $x$ as $(z-x{)}^n$. The function $f(z)=z^2$ then has abstract local expansion $a_x=x^2+2xX+X^2$. To transfer the expansion from $y$ to $x$, substitute $X\mapsto X+x-y$. For example,The expression on the left is the expansion at $y=2$ transported to $x=1$. In particular, ${\Gamma }_{12}{X}^2-X^2=-2X+1$ has strictly lower degree, illustrating the defining triangular property.

A model of a regularity structure adds concrete distributions and analytic bounds to these abstract symbols. Writing their realizations as ${\Pi }_x$, compatibility requires ${\Pi }_x{\Gamma }_{xy}={\Pi }_y$ and ${\Gamma }_{xy}{\Gamma }_{yz}={\Gamma }_{xz}$. For singular equations the symbols extend beyond monomials; constructing a model, proving reconstruction and controlling corrections are additional steps. The triple alone does not make arbitrary products of distributions meaningful.^[ref]