The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation

This setting fixes the notation for the Wick-square problem on the Sobolev triple of order two. It introduces no new concept and asserts nothing beyond the facts recorded below, each carried by the reference attached to it.

1. (The problem)¶ We work in the setting of The Wick-Square Problem on the Torus: Standing Notation: $1\le n\le3$, the modes, weights $\mu_{k}$, cubes $\Gamma_{N}$, the state space $H^{-1}$, the unit families $e_{k}$ and the parameters $\nu,\beta,\gamma$ are those fixed there.

2. (The triple)¶ $(H,V,A)$ is the Sobolev triple of order $2$: $H=H^{-3}(\mathbb{T}^{n})$ and $V=H^{-2}(\mathbb{T}^{n})$. By that clause it is a Hilbert triple with $(V,d_{V})$ separable, and we work in the setting of Hilbert Triples: Standing Notation and Background for it, whose notation is in force: the norms, inner products and distances with subscripts $H$ and $V$, the domain $D(A)$, the penalty function $h(x)=\tfrac12|x|_{V}^{2}$, the classes $C^{2}(U)$, the spaces $\mathrm{Sym}(H)$ and $\mathrm{Sym}(V)$ with the restriction $Y|_{V}$ and the identity forms $I_{H}$, $I_{V}$. By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §domain, $D(A)=H^{-1}$ and $Ax$ is the family $k\mapsto\mu_{k}x(k)$; by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §infinite, $H$ is not finite-dimensional.

3. (Noise and gradient form)¶ $\kappa$ is a fixed enumeration of $\mathbb{Z}^{n}$ as in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families; $f$ denotes the sequence $(e_{\kappa(j)})_{j\in\mathbb{N}}$, square-summable in $V$ with trace $\mathrm{Tr}_{f}$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §noise; and $G\in\mathrm{Sym}(H)$ denotes the form of The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form, with $G(p,p')=\sum_{k\in\mathbb{Z}^{n}}\langle p,e_{k}\rangle_{H}\langle p',e_{k}\rangle_{H}$.

4. (Riccati data)¶ $q_{k}$ and $a_{k}$ are the Riccati coefficients and renormalised constants, and $B_{q}:V\to H$ denotes the map of The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §drift, $(B_{q}x)(k)=2q_{k}x(k)$.

5. (The free solution)¶ $u_{0}:H^{-1}\to\mathbb{R}$ denotes the function $R$ of The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge §limit, $u_{0}(x)=\sum_{k\in\mathbb{Z}^{n}}\bigl(q_{k}x(k)^{2}+a_{k}\bigr)$; the coefficients $q_{k}$ and $a_{k}$ do not involve a running cost.

6. (Background)¶ Regular functions are those of Regular Functions for the Wick-Square Problem: Diagonal Quadratics of Curvature of Order the Inverse Squared Weight, plus Twice Differentiable Functions on the Sobolev Space of Order -3. The results The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift, Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations and Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space are in force by reference.