The Second-Order Structure Condition at Uniquely Mapped Pairs on the Wasserstein Space

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. That a pair of members of $\mathcal{S}(d)$ is admitted at $\alpha$ is the condition of that clause, which concerns matrices only. For an ordered pair $(\mu,\nu)$ of elements of $\mathcal{P}_{2}(\mathbb{R}^{d})$ that is uniquely mapped, with an optimal map $S$ from $\mu$ to $\nu$, the class $\mathrm{id}-S$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (i) and does not depend on the choice of $S$ by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (iii). Applied to the ordered pair $(\nu,\mu)$ when it too is uniquely mapped, with an optimal map $S'$ from $\nu$ to $\mu$, the same two clauses place the class $\mathrm{id}-S'$ in $L^{2}(\nu;\mathbb{R}^{d})$, independently of the choice of $S'$; we write $S'-\mathrm{id}=-(\mathrm{id}-S')$, an element of the vector space $L^{2}(\nu;\mathbb{R}^{d})$. $\alpha^{-1}$ denotes the multiplicative inverse of a positive $\alpha\in\mathbb{R}$, and $|s|$ 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. (Second-order structure pair at a level)(i) 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

$$-\omega_{1}\bigl(\alpha\,W_{2}(\mu,\nu)^{2}+\alpha^{-1}\bigr)-\omega_{2}\bigl(\delta\,(|\mathcal{E}(\mu)|+|\mathcal{E}(\nu)|+1),\,\alpha\bigr)\ \le\ F^{-}_{\delta}\bigl(\mu,r,\alpha(\mathrm{id}-S),\mathbb{X}\bigr)-F^{+}_{\delta}\bigl(\nu,r,\alpha(S'-\mathrm{id}),\mathbb{Y}\bigr)$$

whenever $\alpha,\delta\in\mathbb{R}$ satisfy $1<\alpha$ and $0<\delta<1$; $\mu,\nu\in\mathcal{D}_{\Sigma}$ are such that both ordered pairs $(\mu,\nu)$ and $(\nu,\mu)$ are uniquely mapped, $S$ is an optimal map from $\mu$ to $\nu$ and $S'$ an optimal map from $\nu$ to $\mu$, and

$$\delta\,\bigl(|\mathcal{E}(\mu)|+|\mathcal{E}(\nu)|\bigr)\ \le\ R ;$$

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

2. (The second-order structure condition)(ii) The operator $F$ satisfies the second-order structure condition at uniquely mapped pairs if for every positive $R\in\mathbb{R}$ there is a second-order structure pair for $F$ at $R$.