Second-Order Equation Operators on the Lift of the Wasserstein Space and Their Delta-Shifts

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$. The space $L^{2}(\Omega;\mathbb{R}^{d})$ of classes of square-integrable random vectors and the law $\mathcal{L}(X)$ of a class are those of that clause, the set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices is that of that clause, the joint law $\mathcal{L}(X,V)$ of two classes is that of that clause, for $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ the preimage $Q^{\Lambda}$ of $Q$ under the law map is that of that clause, and products of sets are Cartesian products. In this definition the letter $r$ denotes a real number, $V$ a class in $L^{2}(\Omega;\mathbb{R}^{d})$ and $\mathbb{X}$ an element of $\mathcal{S}(d)$; the dimension written $r$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation is not used.

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

$$F:\ Q^{\Lambda}\times\mathbb{R}\times L^{2}(\Omega;\mathbb{R}^{d})\times\mathcal{S}(d)\to\mathbb{R},$$

whose value at $(X,r,V,\mathbb{X})$ is written $F(X,r,V,\mathbb{X})$, with the following property. (Law-invariance in the pair)(ii) For all $X,X'\in Q^{\Lambda}$, all $V,V'\in L^{2}(\Omega;\mathbb{R}^{d})$, every $r\in\mathbb{R}$ and every $\mathbb{X}\in\mathcal{S}(d)$,

$$\mathcal{L}(X,V)=\mathcal{L}(X',V')\quad\text{implies}\quad F(X,r,V,\mathbb{X})=F(X',r,V',\mathbb{X}).$$

2. (The $\delta$-shifts of $F$)(iii) Let $F$ be a second-order equation operator on the lift over $\mathcal{D}_{\Sigma}$ and let $\delta\in\mathbb{R}$ be positive. For $X\in\mathcal{D}_{\Sigma}^{\Lambda}$ write $\nu=\mathcal{L}(X)$, an element of $\mathcal{D}_{\Sigma}$ and hence of $\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i), so that $\mathcal{E}(\nu)$ is a real number and the score $\Sigma(\nu)$ lies in $T_{\nu}$, hence in the space $L^{2}(\nu;\mathbb{R}^{d})$, so that the composition $\Sigma(\nu)\circ X$ is an element of $L^{2}(\Omega;\mathbb{R}^{d})$. The $\delta$-shifts of $F$ relative to the penalty pair are the two functions

$$F^{-}_{\delta},\ F^{+}_{\delta}:\ \mathcal{D}_{\Sigma}^{\Lambda}\times\mathbb{R}\times L^{2}(\Omega;\mathbb{R}^{d})\times\mathcal{S}(d)\to\mathbb{R}$$

given by

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

Both are second-order equation operators on the lift over $\mathcal{D}_{\Sigma}$. Indeed, let $X'\in\mathcal{D}_{\Sigma}^{\Lambda}$ and $V'\in L^{2}(\Omega;\mathbb{R}^{d})$ satisfy $\mathcal{L}(X',V')=\mathcal{L}(X,V)$. By The Joint Law of Two Classes of Square-Integrable Random Vectors: Marginals, Cost, and Invariance Under Shifts by a Vector Field (iii), $\mathcal{L}(X')=\nu$, so the shifts at $X'$ are formed with the same $\mathcal{E}(\nu)$ and $\Sigma(\nu)$ as at $X$, and, by the same clause with $\eta=\Sigma(\nu)$ and $t=\delta$, respectively $t=-\delta$, $\mathcal{L}(X,V+\delta\,\Sigma(\nu)\circ X)=\mathcal{L}(X',V'+\delta\,\Sigma(\nu)\circ X')$ and $\mathcal{L}(X,V-\delta\,\Sigma(\nu)\circ X)=\mathcal{L}(X',V'-\delta\,\Sigma(\nu)\circ X')$; hence $F^{-}_{\delta}(X,r,V,\mathbb{X})=F^{-}_{\delta}(X',r,V',\mathbb{X})$ and $F^{+}_{\delta}(X,r,V,\mathbb{X})=F^{+}_{\delta}(X',r,V',\mathbb{X})$ by the law-invariance of $F$.