= Polynomial model of a regularity structure
{title2=$(\Pi_xX^k)(z)=(z-x)^k$}
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 $X_1,\ldots,X_d$, graded by total degree. For a <multi-index> $k=(k_1,\ldots,k_d)$,
$$
X^k=X_1^{k_1}\cdots X_d^{k_d},\qquad |k|=k_1+\cdots+k_d,\qquad
(\Pi_xX^k)(z)=(z-x)^k.
$$
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>.<A theory of regularity structures by Martin Hairer>[§2.2, polynomial regularity structure; §2.3, Definition 2.17 and Remark 2.23, canonical polynomial model]{ref}
For example, $f(z_1,z_2)=z_1z_2$ has local abstract expansion $a_x=2+2X_1+X_2+X_1X_2$ at $x=(1,2)$. At $y=(3,4)$ it has $a_y=12+4X_1+3X_2+X_1X_2$. Transporting the latter substitutes $X_1-2,X_2-2$:
$$
\Gamma_{xy}a_y=12+4(X_1-2)+3(X_2-2)+(X_1-2)(X_2-2)
=2+2X_1+X_2+X_1X_2=a_x.
$$
The coefficients change because the expansion point changes; the represented <function> is still $z_1z_2$.
The analytic scaling also follows by calculation. For a smooth <test function> and $\varphi_x^\lambda(z)=\lambda^{-d}\varphi((z-x)/\lambda)$, changing variables gives
$$
\langle\Pi_xX^k,\varphi_x^\lambda\rangle
=\lambda^{|k|}\int_{\mathbb R^d}w^k\varphi(w)\,dw.
$$
Thus 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>.
Back to article page
