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

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $(H,V,A)$ be the Sobolev triple of order $2$, so that $H=H^{-3}(\mathbb{T}^{n})$ and $V=H^{-2}(\mathbb{T}^{n})$, linear subspaces of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ carrying the inner products, norms and distances of orders $-3$ and $-2$, written with the subscripts $H$ and $V$; $D(A)\subseteq V$ and $A:D(A)\to H$ are the domain and the form operator of the triple. Since $(V,d_{V})$ is separable by The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian §triple, we may and do work in the setting of Hilbert Triples: Standing Notation and Background for this triple. $\mathrm{Sym}(H)$ and $\mathrm{Sym}(V)$ are the sets of bounded symmetric bilinear forms on $H$ and on $V$, with the orders $\preceq$, the zero form $0_{\mathrm{Sym}}$ and the identity forms $I_{H}$ and $I_{V}$ fixed there. The unit families $e_{k}$ of The Wick-Square Problem on the Torus: Standing Notation §units lie in $H^{-1}$, hence in $V$ and in $H$ (The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §inclusion). Let $\kappa$ be an enumeration of $\mathbb{Z}^{n}$ as in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families, and let $q_{k}$ be the Riccati coefficients. Then the following hold.

1. (The domain)¶ $D(A)=H^{-1}$, and for every $x\in H^{-1}$ the vector $Ax$ is the family $k\mapsto\mu_{k}x(k)$ and $|Ax|_{H}=|x|_{H^{-1}}$.

2. (Pairings)¶ For $z\in H$ and $k\in\mathbb{Z}^{n}$, $\langle z,e_{k}\rangle_{H}=z(k)/\mu_{k}^{3}$; for $z\in V$, $\langle z,e_{k}\rangle_{V}=z(k)/\mu_{k}^{2}$. For $x\in H^{-1}$ and $z\in H$ the families $k\mapsto|x(k)z(k)|/\mu_{k}^{2}$ and $k\mapsto x(k)z(k)/\mu_{k}^{2}$ are cube-summable and $\langle Ax,z\rangle_{H}=\sum_{k\in\mathbb{Z}^{n}}x(k)z(k)/\mu_{k}^{2}$.

3. (Infinite dimension)¶ $H$, as a vector space over $\mathbb{R}$, is not finite-dimensional.

4. (White noise)¶ The sequence $f=(e_{\kappa(j)})_{j\in\mathbb{N}}$ is square-summable in $V$, with $\sigma(f)=\sum_{k\in\mathbb{Z}^{n}}1/\mu_{k}^{2}$; and for every $X\in\mathrm{Sym}(V)$ the family $k\mapsto X(e_{k},e_{k})$ is cube-summable, with lattice sum the trace $\mathrm{Tr}_{f}X$.

5. (The gradient form)¶ For all $p,p'\in H$ the families $k\mapsto|\langle p,e_{k}\rangle_{H}\langle p',e_{k}\rangle_{H}|$ and $k\mapsto\langle p,e_{k}\rangle_{H}\langle p',e_{k}\rangle_{H}$ are cube-summable, and the map $G:H\times H\to\mathbb{R}$ sending $(p,p')$ to the lattice sum of the latter belongs to $\mathrm{Sym}(H)$ and satisfies $0_{\mathrm{Sym}}\preceq G\preceq I_{H}$.

6. (The Riccati drift)¶ For every $x\in V$ the family $B_{q}x:k\mapsto2q_{k}x(k)$ lies in $H$, and the map $B_{q}:V\to H$ so defined is a monotone nonlinearity for $(H,V,A)$.