The Second-Order Structure Condition at Optimally Coupled Pairs 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 $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair.

The space $L^{2}(\Omega;\mathbb{R}^{d})$ with its norm $\lVert\cdot\rVert_{L^{2}}$, its differences and its scalar multiples, and the law $\mathcal{L}(X)$ of a class, are those of that clause, and $\mathcal{D}_{\Sigma}^{\Lambda}$ is the preimage of $\mathcal{D}_{\Sigma}$ under the law map, so that $\mathcal{L}(X)\in\mathcal{D}_{\Sigma}$ for $X\in\mathcal{D}_{\Sigma}^{\Lambda}$; since $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i), the penalty $\mathcal{E}(\mathcal{L}(X))$ is then a real number. The set $\mathcal{S}(d)$ with its norm $\lVert\cdot\rVert$ and its ordering $\preceq$ is that of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iii), and the dot product $z\cdot w$ of points of $\mathbb{R}^{d}$ and the matrix-vector product $Az$ are those of Euclidean Space and Real Matrices: Standing Notation (i) and Euclidean Space and Real Matrices: Standing Notation (ii), whose notation Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation puts in force. That a pair $(X,Y)$ of elements of $L^{2}(\Omega;\mathbb{R}^{d})$ is optimally coupled is as defined there. We write $3=2+1$, $6=3+3$ and $\lVert Z\rVert_{L^{2}}^{2}=\lVert Z\rVert_{L^{2}}\lVert Z\rVert_{L^{2}}$, and $\alpha^{-1}$ denotes the multiplicative inverse of a positive $\alpha\in\mathbb{R}$; $|s|$ 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.

1. (Pairs admitted at a doubling strength)(i) Let $\alpha\in\mathbb{R}$ be positive. A pair $(\mathbb{X},\mathbb{Y})$ of members of $\mathcal{S}(d)$ is admitted at $\alpha$ if $\mathbb{X}\preceq\mathbb{Y}$, $\lVert\mathbb{X}\rVert\le6\alpha$, $\lVert\mathbb{Y}\rVert\le6\alpha$, and

$$-3\alpha\bigl(\lVert z\rVert^{2}+\lVert w\rVert^{2}\bigr)\ \le\ z\cdot(\mathbb{X}z)-w\cdot(\mathbb{Y}w)\ \le\ 3\alpha\lVert z-w\rVert^{2}\qquad\text{for all }z,w\in\mathbb{R}^{d}.$$

2. (Second-order structure pair at a level)(ii) Let $R\in\mathbb{R}$ be positive, let $\omega_{1}$ be a modulus of continuity, and let $\omega_{2}$ be a function with values in $\mathbb{R}$ on the set of all pairs $(t,\alpha)$ of real numbers with $0\le t$ and $1<\alpha$, such that for every real $\alpha>1$ the function on the set of nonnegative reals with value $\omega_{2}(t,\alpha)$ at $t$ is a modulus of continuity. We say that $(\omega_{1},\omega_{2})$ is a second-order structure pair for $F$ at $R$ if

$$-\omega_{1}\bigl(\alpha\lVert X-Y\rVert_{L^{2}}^{2}+\alpha^{-1}\bigr)-\omega_{2}\bigl(\delta\,(|\mathcal{E}(\mathcal{L}(X))|+|\mathcal{E}(\mathcal{L}(Y))|+1),\,\alpha\bigr)\ \le\ F^{-}_{\delta}\bigl(X,r,\alpha(X-Y),\mathbb{X}\bigr)-F^{+}_{\delta}\bigl(Y,r,\alpha(X-Y),\mathbb{Y}\bigr)$$

whenever $\alpha,\delta\in\mathbb{R}$ satisfy $1<\alpha$ and $0<\delta<1$, the classes $X,Y\in\mathcal{D}_{\Sigma}^{\Lambda}$ are such that $(X,Y)$ is optimally coupled and

$$\delta\,\bigl(|\mathcal{E}(\mathcal{L}(X))|+|\mathcal{E}(\mathcal{L}(Y))|\bigr)\ \le\ R ,$$

$r\in\mathbb{R}$ satisfies $-R\le r\le R$, and $(\mathbb{X},\mathbb{Y})$ is a pair admitted at $\alpha$ in the sense of clause 1. The arguments of the two moduli are nonnegative: $\alpha\lVert X-Y\rVert_{L^{2}}^{2}+\alpha^{-1}$ is the sum of a product of nonnegative reals and a positive multiplicative inverse, and $\delta\,(|\mathcal{E}(\mathcal{L}(X))|+|\mathcal{E}(\mathcal{L}(Y))|+1)$ is the product of the positive real $\delta$ with a sum of two nonnegative absolute values and $1$, hence positive.

3. (The second-order structure condition)(iii) The operator $F$ satisfies the second-order structure condition at optimally coupled pairs if for every positive $R\in\mathbb{R}$ there is a second-order structure pair for $F$ at $R$.