First-Order Equation Operators on the Noise Wasserstein Space with Momentum-Continuous Shifts

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ and let $F$ be a first-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 $T^{a}_{\nu}$, hence in the real Hilbert space $L^{2}(\nu;X^{a})$, by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair; for $q,q'\in L^{2}(\nu;X^{a})$ the difference $q-q'$ is taken in that space; and $|s|$ is the absolute value of $s\in\mathbb{R}$.

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

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

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