The First-Order Structure Condition at Uniquely Noise-Mapped Pairs

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 $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ and a noise-optimal map $S$ from $\mu$ to $\nu$, the displacement $S-\mathrm{id}$ is an element of $L^{2}(\mu;X^{a})$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, and we write $\mathrm{id}-S=-(S-\mathrm{id})$ in that vector space. $\alpha^{-1}$ denotes the multiplicative inverse of a positive $\alpha\in\mathbb{R}$, and $|s|$ the absolute value of $s\in\mathbb{R}$.

1. (Structure pair at a level)¶ 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 structure pair for $F$ at $R$ if

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

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 noise-mapped, $S$ is a noise-optimal map from $\mu$ to $\nu$ and $S'$ a noise-optimal map from $\nu$ to $\mu$, and

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

and $r\in\mathbb{R}$ satisfies $-R\le r\le R$. Here $W_{a}(\mu,\nu)$ is a nonnegative real number and $\mathcal{E}(\mu)$, $\mathcal{E}(\nu)$ are defined, because $\mu,\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair and by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected and The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric. The arguments of the two moduli are nonnegative: $\alpha\,W_{a}(\mu,\nu)^{2}+\alpha^{-1}$ 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 structure condition)¶ The operator $F$ satisfies the first-order structure condition at uniquely noise-mapped pairs if for every positive $R\in\mathbb{R}$ there is a structure pair for $F$ at $R$.