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]

A model of a regularity structure realizes the abstract symbols of a regularity structure $(A,T,G)$ as concrete distributions, consistently across expansion points. It comprises continuous linear realization maps ${\Pi }_x:T\to {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ and maps ${\Gamma }_{xy}\in G$ satisfyingThus ${\Gamma }_{xy}$ transports an expansion from $y$ to $x$ without changing the represented distribution.^[ref]

Analytic bounds connect each symbol's assigned order $\alpha$ to its actual small-scale size. For ordinary isotropic scaling, put ${\phi }_x^{\lambda }(z)={\lambda }^{-d}\phi ((z-x)/\lambda )$. For $\tau \in T_{\alpha }$ one requiresThese bounds are uniform on each compact set of base points, below each fixed upper order, for $0<\lambda \le 1$ and smooth test functions supported in the unit ball with $C^r$ norm at most one; choose an integer $r>\max (0,-\inf A)$. Other scaling choices change the rescaled test functions and distance. Negative $\alpha$ allows singular distributions, whose small-scale averages can grow as $\lambda$ shrinks.^[ref]

In the one-dimensional polynomial model of a regularity structure, $({\Pi }_x{X}^n)(z)=(z-x{)}^n$ and ${\Gamma }_{xy}X=X+x-y$. For $x=1,y=2$,A change of variable $z=x+\lambda w$ in the test function pairing gives exactly ${\lambda }^n\int w^n\phi (w)dw$, explaining the analytic bound for the monomial $X^n$. The realization maps supply information absent from the abstract triple alone.

The polynomial model of a regularity structure realizes abstract monomials as ordinary coordinate differences around each base point. It is the example that turns a regularity structure into familiar Taylor polynomials. In $d$ dimensions, use a polynomial ring in symbols $X_1,\dots ,X_d$, graded by total degree. For a multi-index $k=(k_1,\dots ,k_d)$,Re-expansion uses ${\Gamma }_{xy}p(X)=p(X+x-y)$. Substitution immediately gives ${\Pi }_x{\Gamma }_{xy}={\Pi }_y$ and ${\Gamma }_{xy}{\Gamma }_{yz}={\Gamma }_{xz}$, the compatibility identities of a model of a regularity structure.^[ref]

For example, $f(z_1,z_2)=z_1{z}_2$ has local abstract expansion $a_x=2+2X_1+X_2+X_1{X}_2$ at $x=(1,2)$. At $y=(3,4)$ it has $a_y=12+4X_1+3X_2+X_1{X}_2$. Transporting the latter substitutes $X_1-2,X_2-2$:The coefficients change because the expansion point changes; the represented function is still $z_1{z}_2$.

The analytic scaling also follows by calculation. For a smooth test function and ${\phi }_x^{\lambda }(z)={\lambda }^{-d}\phi ((z-x)/\lambda )$, changing variables givesThus the assigned degree predicts exactly the power of $\lambda$ in the model bound. Anisotropic scaling can assign different weights to different coordinates instead; ordinary total degree is the isotropic choice. General regularity structures retain this organization of local expansions while admitting symbols that are not polynomials.