A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation

This setting fixes the standing notation for working relative to a diagonal Gaussian reference measure on the noise Wasserstein space, through rescaled heads and Gaussian tails. 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 and Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation are in force: the space $X$ with basis $(e_{k})_{k\in\mathbb{N}}$, coordinates $x_{k}$ and the maps $p_{n}$, $p_{n}^{*}$, $P_{n}$ and $Q_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates; 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 $L^{2}(\mu;X^{a})$ of that setting. $\mathcal{P}^{a}_{\rho}$ is the set of The Measures Noise-Connected to the Reference Measure §space and $W_{a}$ the noise Wasserstein distance, a metric on $\mathcal{P}^{a}_{\rho}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric.

2. (The Gaussian reference measure)¶ $c$ is a variance sequence, with truncations $c^{(n)}$ and diagonal Gaussian measures $\gamma_{c^{(n)}}$ on $\mathbb{R}^{n}$; $\gamma_{c}$ is the diagonal Gaussian measure on $X$ with variances $c$; and the reference measure is $\rho=\gamma_{c}$, which is admissible since $\gamma_{c}\in\mathcal{P}_{2}(X)$ by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §moment. Finite relative entropy and $H(\cdot\,|\,\cdot)$ are those of that definition, on $(X,\mathcal{B}(X))$ and on $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$. The relative score with respect to $\gamma_{c}$ and its components are those of that definition, with $L^{2}(\mu)$ as there; $a$ is a weight sequence by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights, and finite Fisher information relative to $\gamma_{c}$ with weights $a$ and its value $\mathcal{I}_{a}(\cdot\,|\,\gamma_{c})$ are those of Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information.

3. (Rescaled heads)¶ For $n\in\mathbb{N}$, the rescaled head map $r_{n}:X\to\mathbb{R}^{n}$, the numbers $a_{k}^{1/2}$ and $a_{k}^{-1/2}$, and the lift $\Lambda_{n}\eta$ of a Borel map $\eta:\mathbb{R}^{n}\to\mathbb{R}^{n}$ are those of Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space, whose claims are in force. $\delta_{n}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the map $\delta_{n}(u)=(a_{1}^{-1/2}u_{1},\dots,a_{n}^{-1/2}u_{n})$ (that is, $r_{n}=\delta_{n}\circ p_{n}$). For $\mu\in\mathcal{P}(X)$, $\tilde{\mu}_{n}=(r_{n})_{\#}\mu$. The vector $\tilde{c}^{(n)}=(c_{1}/a_{1},\dots,c_{n}/a_{n})$ is a variance vector, each entry being a quotient of positive numbers; $\tilde{\gamma}_{n}=\gamma_{\tilde{c}^{(n)}}$ is the diagonal Gaussian measure on $\mathbb{R}^{n}$ with variances $\tilde{c}^{(n)}$, and $\rho_{\tilde{c}^{(n)}}$ and $|u|_{\tilde{c}^{(n)}}^{2}$ are the diagonal Gaussian density and the weighted square of that definition.

4. (Gaussian tails)¶ For $n\in\mathbb{N}$, $\tau_{n}=(Q_{n})_{\#}\gamma_{c}\in\mathcal{P}(X)$, the map $Q_{n}$ being Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity. The product $\mathbb{R}^{n}\times X$ carries the product metric; its Borel $\sigma$-algebra is $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ by The Borel Sigma-Algebra of a Product of Two Separable Metric Spaces is the Product Sigma-Algebra §product, both factors being separable by Euclidean Space is a Separable Metric Space §separable and Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space; product measures on it are those of Existence and Uniqueness of the Product Measure. $\Psi_{n}:\mathbb{R}^{n}\times X\to X$ is the map

$$\Psi_{n}(u,w)=\sum_{k=1}^{n}a_{k}^{1/2}u_{k}\,e_{k}+w .$$

5. (Euclidean heads)¶ The Euclidean items cited for measures on $\mathbb{R}^{n}$ are read with $n$ in place of the dimension written $d$ there, in the settings named there: the set $\mathcal{P}(\mathbb{R}^{n})$ of Borel probability measures and push-forwards of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, the set $\mathcal{P}_{2}(\mathbb{R}^{n})$ of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, the quadratic Wasserstein distance $W_{2}$, optimal couplings, the spaces of square-integrable vector fields $L^{2}(\lambda;\mathbb{R}^{n})$ and the real Hilbert spaces $L^{2}(\lambda)=L^{2}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda)$ of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert, the tangent spaces $T_{\lambda}$, and, for $\lambda\in\mathcal{P}_{2}(\mathbb{R}^{n})$, the Ornstein-Uhlenbeck functional $\ell^{\tilde{c}^{(n)}}_{\lambda}$, finite Fisher information relative to $\tilde{\gamma}_{n}$, the relative score $\zeta^{\tilde{c}^{(n)}}_{\lambda}\in T_{\lambda}$ with its components $(\zeta^{\tilde{c}^{(n)}}_{\lambda})_{k}\in L^{2}(\lambda)$ of Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded C^1 Functions, and the relative Fisher information $\mathcal{I}(\lambda\,|\,\tilde{\gamma}_{n})$.