The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space; for $\nu\in\mathcal{P}^{a}_{\rho}$ let $L^{2}(\nu;X^{a})$ be the real Hilbert space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields and $T^{a}_{\nu}\subseteq L^{2}(\nu;X^{a})$ the noise tangent space, and let products of sets be Cartesian products. In this definition the letter $q$ denotes a noise field and the letter $r$ a real number.

1. (Noise fields over a set of measures)¶ For a subset $Q\subseteq\mathcal{P}^{a}_{\rho}$,

$$\mathcal{V}^{a}(Q)=\bigl\{\,(\nu,q)\ :\ \nu\in Q,\ q\in L^{2}(\nu;X^{a})\,\bigr\}$$

denotes the set of pairs consisting of a measure in $Q$ and a square-integrable noise field against it, called the bundle of noise fields over $Q$.

2. (First-order equation operator)¶ For a subset $Q\subseteq\mathcal{P}^{a}_{\rho}$, a first-order equation operator over $Q$ is a function

$$F:\ \mathcal{V}^{a}(Q)\times\mathbb{R}\to\mathbb{R},$$

whose value at $((\nu,q),r)$ is written $F(\nu,r,q)$.

3. (The $\delta$-shifts of $F$)¶ Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$, let $F$ be a first-order equation operator over $\mathcal{D}_{\Sigma}$, and let $\delta\in\mathbb{R}$ satisfy $0<\delta$. For $\nu\in\mathcal{D}_{\Sigma}$ one has $\nu\in\mathcal{D}$, so the real number $\mathcal{E}(\nu)$ is defined, and the score $\Sigma(\nu)$ lies in $T^{a}_{\nu}$, hence in $L^{2}(\nu;X^{a})$, both by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair; so for $q\in L^{2}(\nu;X^{a})$ the fields $q+\delta\,\Sigma(\nu)$ and $q-\delta\,\Sigma(\nu)$ belong to the vector space $L^{2}(\nu;X^{a})$. The $\delta$-shifts of $F$ relative to the noise penalty pair are the two functions

$$F^{-}_{\delta},\ F^{+}_{\delta}:\ \mathcal{V}^{a}(\mathcal{D}_{\Sigma})\times\mathbb{R}\to\mathbb{R}$$

given by

$$F^{-}_{\delta}(\nu,r,q)=F\bigl(\nu,\ r+\delta\,\mathcal{E}(\nu),\ q+\delta\,\Sigma(\nu)\bigr),\qquad F^{+}_{\delta}(\nu,r,q)=F\bigl(\nu,\ r-\delta\,\mathcal{E}(\nu),\ q-\delta\,\Sigma(\nu)\bigr).$$