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  "body": "\n## Definition\n\nA *regularity structure* is a triple $\\mathscr{T} = (A, T, G)$ such that:\n\n1. $A \\subset \\R$ is an index set that is bounded below, locally finite, and contains $0$.\n2. $T = \\bigoplus_{\\alpha \\in A} T_\\alpha$ is a graded vector space, each $T_\\alpha$ is a [[Banach space|06GZ956657]], and $T_0 \\cong \\R$ is spanned by an element written $\\mathbf{1}$.\n3. $G$ is a [[group|G5S1M9]] of continuous [[linear maps|0NRYHJPQG2]] $T \\to T$ such that for all $\\Gamma \\in G$, $\\alpha \\in A$ and $\\tau \\in T_\\alpha$,\n$$\\Gamma \\tau - \\tau \\in \\bigoplus_{\\beta \\in A, \\ \\beta \u003c \\alpha} T_\\beta,$$\nand $\\Gamma \\mathbf{1} = \\mathbf{1}$.\n\nThe set $A$ is the set of homogeneities, $T$ is the model space, and $G$ is the structure group.\n\n## Remarks\n\nExample: the polynomial regularity structure on $\\R^d$.\nTake $A = \\N$, 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 $(\\Gamma_h P)(X) = P(X + h)$.\nThe 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 $\\deg P$.\nNon-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 $1$.\nTo use a structure one adds a model, which realises each symbol as a concrete distribution near each point.\nThe reconstruction theorem then turns local expansions into a global distribution.\n",
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