The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation

This setting fixes the standing notation for working on the space of distributions on the torus relative to the free field, with displacements measured in the norm of square-integrable white noise. It introduces no new concept and asserts nothing beyond what the references attached to it supply.

1. (The torus)¶ The notation of The Flat Torus: Standing Notation is in force with a natural number $n$ satisfying $1\le n$, the dimension of the torus $\mathbb{T}^{n}$; the letter $\pi$ always denotes the real number of The Number Pi §pi (the wrapping map of The Flat Torus: Standing Notation §cell is not used, and the couplings and coordinate maps written $\pi$, $\pi_{1}$, $\pi_{2}$ in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation are not written in items adopting this setting), and the trigonometric system of the torus is used only through its classes $E_{k}$. $L^{2}(\mathbb{T}^{n})$ carries the inner product $\langle\cdot,\cdot\rangle_{L^{2}}$ and norm $\lVert\cdot\rVert_{L^{2}}$. Coefficient families, the Fourier coefficient family $\hat{U}$ of a class $U\in L^{2}(\mathbb{T}^{n})$, the classes $E_{k}$, enumerations of $\mathbb{Z}^{n}$, the Fourier weights $\mu_{k}=1+4\pi^{2}\lVert k\rVert^{2}$ and the numbers $\rho_{k}$ are those of Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families. $\Delta$ is the Laplacian of a map of class $C^{2}$ on $\mathbb{R}^{n}$.

2. (The data)¶ $m\in\mathbb{N}$ satisfies $n\le m+1$, $\kappa$ is a fixed enumeration of $\mathbb{Z}^{n}$, $X=H^{-m}(\mathbb{T}^{n})$ is the Sobolev space of order $-m$, a real Hilbert space with the inner product $\langle\cdot,\cdot\rangle_{H^{-m}}$ 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, and $e_{j}$, $a_{j}$ and $c_{j}$ ($j\in\mathbb{N}$) are those of White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space for these $n$, $m$ and $\kappa$. Thus $(e_{j})_{j\in\mathbb{N}}$ is an orthonormal basis of $X$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §basis, $a$ is a weight sequence bounded by $1$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §weights, and $c$ is a variance sequence by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §variances. For $\Phi\in X$ and $k\in\mathbb{Z}^{n}$, $\Phi(k)$ is the value of the coefficient family $\Phi$ at $k$. The letter $\kappa$ always denotes this enumeration; where a cited item requires a positive constant $\kappa$ with $c_{j}\le\kappa\,a_{j}$, it is called the constant of that item.

3. (The Gaussian reference)¶ The notation of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation is in force with the space $X$ and orthonormal basis $(e_{j})_{j\in\mathbb{N}}$ of clause 2, the noise weights $a$ with bound $\bar{a}=1$, and the variance sequence $c$; the reference measure is $\rho=\gamma_{c}$, the diagonal Gaussian measure on $X$ with variances $c$, referred to as the free field. The letter $\rho$ without a subscript always denotes this measure, and $\rho_{k}$ the numbers of clause 1; the letter $n$ always denotes the dimension of the torus, so the coordinate maps $p_{N}$ and the other maps of that setting indexed by a natural number are written with indices such as $N$, and items stated with a natural number written $n$ in that setting and in the definitions and lemmas it cites are read with $N$ in place of that $n$. The noise space $X^{a}$ and its norm $|\cdot|_{a}$ are described by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §noise-space.