Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation

This setting fixes two readings of the standing notation of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation and First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, called the fine and the coarse level, and the level-indexed notation used by items comparing them. It introduces no new concepts and asserts nothing beyond what the references attached to it supply.

1. (Background)¶ The notation and background of The Real Numbers: Standing Notation and Background are in force.

2. (Levels)¶ For each $i\in\{1,2\}$, $X_{i}$ is a real Hilbert space with inner product $\langle\cdot,\cdot\rangle_{i}$, norm $|\cdot|_{i}$ and distance $d_{i}$, $(e^{(i)}_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $X_{i}$, $a^{(i)}=(a^{(i)}_{k})_{k\in\mathbb{N}}$ is a weight sequence, $\bar{a}^{(i)}\in\mathbb{R}$ is a number with $a^{(i)}_{k}\le\bar{a}^{(i)}$ for every $k\in\mathbb{N}$, as that definition provides, and positive, as there by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights, and $\rho_{i}$ is a member of the set $\mathcal{P}_{2}(X_{i})$; these are data of the kind fixed by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space, Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights and Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §reference. Level $i$ refers to the setting Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, together with the setting First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation layered on it, read with $X_{i}$, $\langle\cdot,\cdot\rangle_{i}$, $|\cdot|_{i}$, $d_{i}$, $(e^{(i)}_{k})_{k\in\mathbb{N}}$, $a^{(i)}$, $\bar{a}^{(i)}$ and $\rho_{i}$ in place of $X$, $\langle\cdot,\cdot\rangle$, $|\cdot|$, $d$, $(e_{k})_{k\in\mathbb{N}}$, $a$, $\bar{a}$ and $\rho$; a notion fixed by those settings, or by an item they cite, is said to be taken at level $i$ when it is read in this way. Level 1 is also referred to as the fine level and level 2 as the coarse level. Objects attached to level 1 and to level 2 are kept apart even when $X_{1}$ and $X_{2}$ coincide as sets or all their data coincide: every measure, noise field, function, test function, penalty pair and operator is attached to the level stated where it is introduced, and every notion applied to it is taken at that level.

3. (Coordinates)¶ For $i\in\{1,2\}$ the coordinates and coordinate maps of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates are taken at level $i$: for $x\in X_{i}$ and $k\in\mathbb{N}$, $x_{k}=\langle x,e^{(i)}_{k}\rangle_{i}$ is the $k$-th coordinate of $x$, and for $n\in\mathbb{N}$, $\varpi^{(i)}_{n}:X_{i}\to\mathbb{R}^{n}$ is the coordinate map $\varpi^{(i)}_{n}(x)=(x_{1},\dots,x_{n})$. Wherever an item taken at level $i$, such as Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §space or The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, uses the basis $(e_{k})_{k\in\mathbb{N}}$, the coordinates $x_{k}$ or the coordinate maps $p_{n}$, these are read as $(e^{(i)}_{k})_{k\in\mathbb{N}}$, the coordinates of $x\in X_{i}$ just fixed, and $\varpi^{(i)}_{n}$; the letter $p$ is thus left free for the mode restrictions of items comparing the levels. The coordinate maps $\pi_{1},\pi_{2}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs keep their names at both levels: on $X_{i}\times X_{i}$ they are the coordinate maps onto the first and the second factor $X_{i}$, the subscript naming the factor and not the level. The spans $X_{n}$, the maps $P_{n}$ and $Q_{n}$ and the synthesis maps of that clause are not used: $X_{1}$ and $X_{2}$ always denote the spaces of clause 2.

4. (Level-indexed notation)¶ For $i\in\{1,2\}$, notation taken at level $i$ carries the index $i$. $\mathrm{id}_{i}$ is the identity map of $X_{i}$, and $\mathcal{P}(X_{i})$ the set of Borel probability measures on $X_{i}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures. $X_{i}^{a}$ is the noise space of $a^{(i)}$, with the noise pairing $\langle\cdot,\cdot\rangle_{a,i}$ and noise norm $|\cdot|_{a,i}$. For $\mu\in\mathcal{P}(X_{i})$, $L^{2}(\mu;X_{i}^{a})$ is the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields taken at level $i$, with inner product $\langle\cdot,\cdot\rangle_{\mu,i}$ and norm $\lVert\cdot\rVert_{\mu,i}$. $\Pi^{a,i}(\mu,\nu)$ is the set of couplings of finite noise cost and $I^{a,i}(\pi)$ the noise cost; $\mathcal{P}^{a}_{\rho_{i}}$ is the set of measures noise-connected to $\rho_{i}$ and $W_{a,i}$ the noise Wasserstein distance. $\mathcal{F}C^{1}_{b}(X_{i})$ is the set of bounded $C^{1}$ cylindrical functions on $X_{i}$ and $\nabla_{a,i}\psi$ the noise gradient of such a function $\psi$; $T^{a,i}_{\mu}$ is the noise tangent space at $\mu\in\mathcal{P}(X_{i})$. For a function $\varphi$ on $\mathcal{P}^{a}_{\rho_{i}}$ and $\mu\in\mathcal{P}^{a}_{\rho_{i}}$, $\nabla_{i}\varphi(\mu)$ is the gradient along noise couplings taken at level $i$, and for $Q\subseteq\mathcal{P}^{a}_{\rho_{i}}$, $\mathcal{V}^{a,i}(Q)$ is the bundle of noise fields over $Q$. At each level the letter conventions of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §operators are in force: the letter $q$ denotes a noise field and the letters $r$, $s$ and $c$ denote real numbers unless an item says otherwise.