The Shift-Coercivity Condition for an Equation Operator on the Lift of the Wasserstein Space

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 $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator on the lift over $\mathcal{D}_{\Sigma}$, with the admissible sets $S^{-}_{\delta,R}$ and $S^{+}_{\delta,R}$ of test data, taken relative to this operator and this penalty pair. For $\nu\in\mathcal{D}_{\Sigma}$ the score $\Sigma(\nu)$ lies in $T_{\nu}$, hence in the space $L^{2}(\nu;\mathbb{R}^{d})$ with its norm $\lVert\cdot\rVert_{\nu}$, and $\mathcal{L}(X)\in\mathcal{D}_{\Sigma}$ for $X\in\mathcal{D}_{\Sigma}^{\Lambda}$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iv). 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.

1. (Score bound at a level)(i) Let $\delta,R\in\mathbb{R}$ satisfy $0<\delta<1$ and $0<R$, and let $C\in\mathbb{R}$ be nonnegative. We say that $C$ is a score bound for $F$ at $(\delta,R)$ if every test datum $(X,r,V,\mathbb{X})$ belonging to $S^{-}_{\delta,R}$ or to $S^{+}_{\delta,R}$ satisfies

$$\bigl\lVert\Sigma(\mathcal{L}(X))\bigr\rVert_{\mathcal{L}(X)}\ \le\ C .$$

2. (The shift-coercivity condition)(ii) The operator $F$ satisfies the shift-coercivity condition if for all $\delta,R\in\mathbb{R}$ with $0<\delta<1$ and $0<R$ there is a score bound for $F$ at $(\delta,R)$.