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

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich, which Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (v) does not assume, and 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. 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, that satisfies the shift-coercivity condition and the shift-semicontinuity condition.

The space $L^{2}(\Omega;\mathbb{R}^{d})$ with its norm $\lVert\cdot\rVert_{L^{2}}$, its differences and its scalar multiples, the law $\mathcal{L}(X)$ of a class and the Wasserstein space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ are those of that clause, and $\mathcal{D}^{\Lambda}$ and $\mathcal{D}_{\Sigma}^{\Lambda}$ are the preimages of $\mathcal{D}$ and of $\mathcal{D}_{\Sigma}$ under the law map. 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 that metric space, and that a pair of elements of $L^{2}(\Omega;\mathbb{R}^{d})$ is optimally coupled is as defined there. We write $\lVert Z\rVert_{L^{2}}^{2}=\lVert Z\rVert_{L^{2}}\lVert Z\rVert_{L^{2}}$ and $W_{2}(\mu,\nu)^{2}=W_{2}(\mu,\nu)W_{2}(\mu,\nu)$, $\tfrac{\alpha}{2}$ for the quotient of $\alpha\in\mathbb{R}$ by $2=1+1$, 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 $F$ on the lift and that $v$ is a viscosity supersolution of $F$ on the lift, both relative to the penalty pair. For positive $\delta\in\mathbb{R}$ the $\delta$-envelope $u^{-}_{\delta}$ of $u$ and the $\delta$-envelope $v^{+}_{\delta}$ of $v$ relative to the penalty pair 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 (iv). 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 Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift (i); the set $\mathcal{D}$ is nonempty, containing the nonempty $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i) and Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (ii), so that Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift may be read with any element of $\mathcal{D}$ in the role of the point written $\mu_{0}$ there, and with the present $e_{0}$. Let $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ satisfy $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})=M(\delta,\alpha)$, as that clause provides, and let $\hat{X},\hat{Y}\in\mathcal{D}^{\Lambda}$ satisfy $\mathcal{L}(\hat{X})=\hat{\mu}$, $\mathcal{L}(\hat{Y})=\hat{\nu}$ and $\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}=W_{2}(\hat{\mu},\hat{\nu})$, as Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift (iv) provides. Assume $0\le M(\delta,\alpha)$.

Finally let $B,R\in\mathbb{R}$ satisfy

$$|b|+|b'|+|e_{0}|\le B,\qquad \delta\,\bigl|\mathcal{E}(\hat{\mu})\bigr|\le B,\qquad \delta\,\bigl|\mathcal{E}(\hat{\nu})\bigr|\le B,\qquad 0<2B\le R,$$

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 maximiser)(i) Then $\hat{X}$ and $\hat{Y}$ belong to $\mathcal{D}_{\Sigma}^{\Lambda}$, the pair $(\hat{X},\hat{Y})$ is optimally coupled, and

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