The Wick-Square Problem on the Torus: Standing Notation

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

1. (Dimension)¶ We work in the setting of The Flat Torus: Standing Notation with a natural number $n$ satisfying $1\le n\le3$. The exponent $p$ and the wrapping map of that setting are not used; $\pi$ denotes the real number of The Number Pi §pi.

2. (Modes and weights)¶ The points $k\in\mathbb{Z}^{n}$ of the integer lattice are called modes, and states are coefficient families $x\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$, with value $x(k)$ at the mode $k$. $\mu_{k}=1+4\pi^{2}\lVert k\rVert^{2}$ is the Fourier weight of $k$, as in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families; $1\le\mu_{k}$ by Summability of the Negative Powers of the Fourier Weights of the Torus §product, used with its $s$ taken to be $n$.

3. (Cutoffs)¶ For $N\in\mathbb{N}$, $\Gamma_{N}$ is the cube of modes whose components lie between $-N$ and $N$, a nonempty finite set by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite; the modes kept at cutoff $N$ are those of $\Gamma_{N}$. Cube sums, cube-summability and the lattice sum $\sum_{k\in\mathbb{Z}^{n}}$ are those of Cube Sums of Families on the Integer Lattice, and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families is in force by reference.

4. (State space)¶ $H^{-1}=H^{-1}(\mathbb{T}^{n})$ is the Sobolev space of order $-1$, a linear subspace of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$, with norm $|\cdot|_{H^{-1}}$ and distance $d_{H^{-1}}$; it is a real Hilbert space by 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 §hilbert. By Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §sobolev with $m=1$ (and $\mu_{k}^{1}=\mu_{k}$), a family $x$ lies in $H^{-1}$ exactly when $k\mapsto x(k)^{2}/\mu_{k}$ is cube-summable, and then $|x|_{H^{-1}}^{2}=\sum_{k\in\mathbb{Z}^{n}}x(k)^{2}/\mu_{k}$.

5. (Unit families)¶ For $k\in\mathbb{Z}^{n}$, $e_{k}$ is the family with value $1$ at $k$ and $0$ at every other mode, as in Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums; it lies in $H^{-1}$ by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units, so $x+te_{k}\in H^{-1}$ for all $x\in H^{-1}$ and $t\in\mathbb{R}$.

6. (Parameters)¶ $\nu$, $\beta$ and $\gamma$ are positive real numbers: the noise intensity, the strength of the quadratic cost, and the discount rate.