Regularity structures

Remarks

Example: the polynomial regularity structure on $\R^d$. Take $A = \N$, let $T_k$ be the space of homogeneous polynomials of degree $k$ in abstract variables $X = (X_1, \ldots, X_d)$, and let $G = \set{\Gamma_h : h \in \R^d}$ with $(\Gamma_h P)(X) = P(X + h)$. The check: $A$ contains $0$ and is locally finite, $T_0$ is spanned by the constant $\mathbf{1}$, $\Gamma_h \Gamma_{h'} = \Gamma_{h + h'}$, and $P(X + h) - P(X)$ has only terms of degree less than $\deg P$. Non-example: adding the scaling $P(X) \mapsto P(2X)$ to $G$ breaks clause 3, since it sends $X_1$ to $2X_1$ and $2X_1 - X_1 = X_1$ has degree $1$, not less than $1$. To use a structure one adds a model, which realises each symbol as a concrete distribution near each point. The reconstruction theorem then turns local expansions into a global distribution.