The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with closed score along couplings, whose penalty domain $\mathcal{D}$ has the map property. 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 satisfies the shift-coercivity condition and the shift-semicontinuity condition. We write $\tfrac{\alpha}{2}$ for the product of $\alpha\in\mathbb{R}$ with the multiplicative inverse of $2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field) and $\alpha^{-1}$ for the multiplicative inverse of a positive $\alpha\in\mathbb{R}$; $|s|$ is the absolute value of $s\in\mathbb{R}$.

Let $u,v:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{D}$; by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Closed Penalty Pair (i), $u$ has penalty-subordinate growth from above and $v$ from below. Assume that $u$ is a viscosity subsolution of $F$ and that $v$ is a viscosity supersolution of $F$, both relative to the penalty pair. For positive $\delta\in\mathbb{R}$ the $\delta$-envelopes $u^{-}_{\delta}$ of $u$ and $v^{+}_{\delta}$ of $v$, functions on $\mathcal{D}$, are then defined. Fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$, as provided by Basic Properties of a Wasserstein-Coercive Penalty Pair (ii).

Let $\delta,\alpha\in\mathbb{R}$ satisfy $0<\delta<1$ and $1<\alpha$, let $\Psi_{\delta,\alpha}:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ be the function with value

$$\Psi_{\delta,\alpha}(\mu,\nu)=u^{-}_{\delta}(\mu)-v^{+}_{\delta}(\nu)-\tfrac{\alpha}{2}\,W_{2}(\mu,\nu)^{2}$$

at $(\mu,\nu)$, and let $M(\delta,\alpha)$ be the supremum of its values, a real number by Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions (i). Assume $0\le M(\delta,\alpha)$. Let $B,R\in\mathbb{R}$ satisfy

$$|b|+|b'|+|e_{0}|\le B,\qquad 0<2B\le R.$$

Let $\lambda$ be a properness constant for $F$ at $R$, and let $(\omega_{1},\omega_{2})$ be a second-order structure pair for $F$ at $R$.

(The structure estimate at a maximising pair)(i) Then there is $(\rho^{*},\sigma^{*})\in\mathcal{D}_{\Sigma}\times\mathcal{D}_{\Sigma}$ with $\Psi_{\delta,\alpha}(\rho^{*},\sigma^{*})=M(\delta,\alpha)$ and

$$\lambda\,M(\delta,\alpha)\ \le\ \omega_{1}\bigl(\alpha\,W_{2}(\rho^{*},\sigma^{*})^{2}+\alpha^{-1}\bigr)+\omega_{2}\bigl(\delta\,(|\mathcal{E}(\rho^{*})|+|\mathcal{E}(\sigma^{*})|+1),\ \alpha\bigr).$$