Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation

This setting fixes the standing notation for analysis on the Lebesgue spaces of a diagonal Gaussian reference measure on a Hilbert space carrying noise weights. It introduces no new concepts and asserts nothing beyond what the references attached to it supply.

1. (Background)¶ The notation and background of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation are in force: the space $X$ with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, orthonormal basis $(e_{k})_{k\in\mathbb{N}}$, coordinates $x_{k}$ and coordinate maps $p_{n}:X\to\mathbb{R}^{n}$; Borel sets, $\mathcal{P}(X)$, $\mathcal{P}_{2}(X)$ and push-forwards; the noise weights $a$, the noise space $X^{a}$ with norm $|\cdot|_{a}$ and orthonormal basis $(f_{k})_{k\in\mathbb{N}}$, and the spaces $L^{2}(\mu;X^{a})$ with inner product $\langle\cdot,\cdot\rangle_{\mu}$ and norm $\lVert\cdot\rVert_{\mu}$; the noise Wasserstein space $\mathcal{P}^{a}_{\rho}$ and distance $W_{a}$; the variance sequence $c$ and the diagonal Gaussian measure $\gamma_{c}$, which is the reference measure $\rho$; finite relative entropy and $H(\cdot\,|\,\cdot)$; the relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, with value $\mathcal{I}_{a}(\cdot\,|\,\gamma_{c})$. Partial derivatives of functions on $\mathbb{R}^{n}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives with $q=n$, and $\lVert\cdot\rVert$ is the Euclidean norm on $\mathbb{R}^{n}$.

2. (Integers, sums and series from $0$)¶ Natural numbers are identified with their images in $\mathbb{R}$ under the canonical map, and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\subseteq\mathbb{R}$. For $n\in\mathbb{N}$, $n!$ is the factorial of $n$, and $0!=1$; for real $y$ and $j\in\mathbb{N}$, $y^{j}$ is the $j$th power of $y$, and $y^{0}=1$; for $n,j\in\mathbb{N}_{0}$ with $j\le n$, $\binom{n}{j}=\frac{n!}{j!\,(n-j)!}$. For $n\in\mathbb{N}_{0}$ and elements $f(0),\dots,f(n)$ of a real vector space, $\sum_{j=0}^{n}f(j)$ is $f(0)$ if $n=0$ and $f(0)+\sum_{j=1}^{n}f(j)$ if $n\in\mathbb{N}$. For a real sequence $(x_{m})_{m\in\mathbb{N}_{0}}$, the series $\sum_{m=0}^{\infty}x_{m}$ converges when $\sum_{m=1}^{\infty}x_{m}$ converges in the sense of Series of Real Numbers §convergent, with sum $x_{0}+\sum_{m=1}^{\infty}x_{m}$.

3. (Lebesgue spaces)¶ For a real number $p$ with $1\le p$, $L^{p}(\gamma_{c})$ is the Lebesgue space $L^{p}(X,\mathcal{B}(X),\gamma_{c})$, with the norm $\lVert\cdot\rVert_{p}$ of The Lebesgue Space of Power-Integrable Functions §norm and the convention of The Lebesgue Space of Power-Integrable Functions §convention: a Borel function $F:X\to\mathbb{R}$ with $\int_{X}|F|^{p}\,d\gamma_{c}<\infty$ is identified with its class. $L^{2}(\gamma_{c})$ is a real Hilbert space with the inner product $\langle\cdot,\cdot\rangle_{L^{2}(\gamma_{c})}$ of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product; closures, closed linear subspaces and orthogonal projections in it are those of Real Hilbert Spaces: Standing Notation and Background, read with $L^{2}(\gamma_{c})$ in place of the space $H$ there.

4. (Cameron-Martin vectors)¶ $H_{c}$ is the Cameron-Martin space of $c$, with Cameron-Martin square $|h|_{c}^{2}$ for $h\in H_{c}$, and for $h\in X$, $\ell_{h}$ is the Paley-Wiener functional of $h$ relative to $c$; The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square is in force by reference.

5. (Rates)¶ For $k\in\mathbb{N}$, $\theta_{k}$ denotes the positive real number $a_{k}/c_{k}$, and $\theta=(\theta_{k})_{k\in\mathbb{N}}$.

6. (Cylindrical functions)¶ $\mathcal{F}C^{1}_{b}(X)$ and $\mathcal{F}C^{2}_{b}(X)$ are the sets of bounded $C^{1}$ and bounded $C^{2}$ cylindrical functions; $\partial_{k}\varphi$ is the $k$-th partial derivative of a function $\varphi$ differentiable on $X$; and $\nabla_{a}\varphi$ is the noise gradient of $\varphi\in\mathcal{F}C^{1}_{b}(X)$. Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions is in force by reference.