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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, for $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ let $L^{2}(\nu;\mathbb{R}^{d})$ be the space of square-integrable vector fields against $\nu$ and $T_{\nu}\subseteq L^{2}(\nu;\mathbb{R}^{d})$ the tangent space, let $\mathcal{S}(d)$ be the set of symmetric real $d\times d$ matrices, and let products of sets be Cartesian products. In this definition the letter $q$ denotes a vector field; the dimension written $q$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i) and in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i) is not used.

1. (Vector fields over a set of measures)(i) For a subset $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$,

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

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

2. (Second-order equation operator)(ii) For a subset $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$, a second-order equation operator over $Q$ is a function

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

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

3. (The $\delta$-shifts of $F$)(iii) Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $F$ be a second-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 $T_{\nu}$, hence in $L^{2}(\nu;\mathbb{R}^{d})$; so for $q\in L^{2}(\nu;\mathbb{R}^{d})$ the fields $q+\delta\,\Sigma(\nu)$ and $q-\delta\,\Sigma(\nu)$ belong to $L^{2}(\nu;\mathbb{R}^{d})$, a real Hilbert space and in particular a real vector space; and the translation Hessian $H_{\mathcal{E}}(\nu)$ of the penalty lies in $\mathcal{S}(d)$, so for $Y\in\mathcal{S}(d)$ the matrices $Y+\delta\,H_{\mathcal{E}}(\nu)$ and $Y-\delta\,H_{\mathcal{E}}(\nu)$ belong to $\mathcal{S}(d)$ by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure. The $\delta$-shifts of $F$ relative to the penalty pair are the two functions

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

given by

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