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. Upper and lower semicontinuity of a real-valued function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, relative to $\mathcal{P}_{2}(\mathbb{R}^{d})$, are understood in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions. 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 $u,v:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ be such that $u$ is upper semicontinuous, $v$ is lower semicontinuous, and $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Then $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and $v$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, since for $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the radius $1$ witnesses that $b$ lies in the set written $A_{u}(\sigma)$ there and that $b'$ lies in the set written $B_{v}(\sigma)$. 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 $v$ are then defined and satisfy $u^{-}_{\delta}=u-\delta\mathcal{E}$ and $v^{+}_{\delta}=v+\delta\mathcal{E}$ on $\mathcal{D}$, by Basic Properties of a Wasserstein-Coercive Penalty Pair §envelopes. 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 §bounded-below.

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

at $(\mu,\nu)$, and let $M(\delta,\alpha)$ be the supremum of its values, a real number by Existence, Penalty Bounds and the Least Penalty at a Maximiser of the Wasserstein-Doubled Difference §maximiser, read with any element of the nonempty set $\mathcal{D}$ in the role of the point written $\mu_{0}$ there and with the present $e_{0}$. Assume $0\le M(\delta,\alpha)$. Let $B,R\in\mathbb{R}$ satisfy

and $\delta\,|\mathcal{E}(\mu)|\le B$ and $\delta\,|\mathcal{E}(\nu)|\le B$ for every $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ with $\Psi_{\delta,\alpha}(\mu,\nu)=M(\delta,\alpha)$. 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)¶ Then there is $(\rho^{*},\sigma^{*})\in\mathcal{D}_{\Sigma}\times\mathcal{D}_{\Sigma}$ with $\Psi_{\delta,\alpha}(\rho^{*},\sigma^{*})=M(\delta,\alpha)$ and