Equation Operators on the Wasserstein Space with Momentum-Continuous Shifts

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair for each positive $\delta\in\mathbb{R}$. For $\nu\in\mathcal{D}_{\Sigma}$ the score $\Sigma(\nu)$ lies in $T_{\nu}$, hence in the real Hilbert space $L^{2}(\nu;\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair; for $q,q'\in L^{2}(\nu;\mathbb{R}^{d})$ the difference $q-q'$ is taken in that space; $\lVert\cdot\rVert$ is the norm on $\mathcal{S}(d)$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §matrices; and $|s|$ is the absolute value of $s\in\mathbb{R}$. In this definition the letter $r$ denotes a real number; the dimension written $r$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation is not used.

(Momentum-continuous shifts)¶ The operator $F$ has momentum-continuous shifts relative to the penalty pair if for all $\delta,R,\eta\in\mathbb{R}$ with $0<\delta<1$, $0<R$ and $0<\eta$ there is a positive $\rho\in\mathbb{R}$ such that

$$\bigl|F^{-}_{\delta}(\nu,r,q,Y)-F^{-}_{\delta}(\nu,r,q',Y)\bigr|<\eta\qquad\text{and}\qquad\bigl|F^{+}_{\delta}(\nu,r,q,Y)-F^{+}_{\delta}(\nu,r,q',Y)\bigr|<\eta$$

for every $\nu\in\mathcal{D}_{\Sigma}$ with $\lVert\Sigma(\nu)\rVert_{\nu}\le R$, every $r\in\mathbb{R}$ with $|r|\le R$, all $q,q'\in L^{2}(\nu;\mathbb{R}^{d})$ with $\lVert q\rVert_{\nu}\le R$, $\lVert q'\rVert_{\nu}\le R$ and $\lVert q-q'\rVert_{\nu}<\rho$, and every $Y\in\mathcal{S}(d)$ with $\lVert Y\rVert\le R$.