The Doubled Difference on the Lift of a Wasserstein-Coercive Penalty Pair: Bounds, Closed Superlevel Sets and the Least Penalty

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 $\mathcal{P}_{2}(\mathbb{R}^{d})$. The space $L^{2}(\Omega;\mathbb{R}^{d})$ with its norm $\lVert\cdot\rVert_{L^{2}}$, metric $d_{L^{2}}$ and differences, 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 $\mathcal{D}^{\Lambda}$ is the preimage of $\mathcal{D}$ under the law map. Write $L^{2}(\Omega;\mathbb{R}^{d})\times L^{2}(\Omega;\mathbb{R}^{d})$ for the product of the real Hilbert space $L^{2}(\Omega;\mathbb{R}^{d})$ with itself, a real Hilbert space by Properties of the Product of Two Real Inner Product Spaces (vi), with its norm written $|\cdot|$.

Upper and lower semicontinuity of a real-valued function on a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$, relative to that subset, are understood in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, and sequential compactness likewise. Having closed superlevel sets in a metric space is as defined there.

Let $\delta\in\mathbb{R}$ be positive and let $u,v:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be such that $u$ is upper semicontinuous and $v$ is lower semicontinuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ relative to $\mathcal{P}_{2}(\mathbb{R}^{d})$, and such that $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and some $b,b'\in\mathbb{R}$. By Basic Properties of a Wasserstein-Coercive Penalty Pair (iv) the $\delta$-envelope $u^{-}_{\delta}$ of $u$ and the $\delta$-envelope $v^{+}_{\delta}$ of $v$ relative to the penalty pair are defined and satisfy $u^{-}_{\delta}=u-\delta\mathcal{E}$ and $v^{+}_{\delta}=v+\delta\mathcal{E}$ on $\mathcal{D}$. 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 $\hat{u},\hat{v}:\mathcal{D}^{\Lambda}\to\mathbb{R}$ be the functions with values $\hat{u}(X)=u^{-}_{\delta}(\mathcal{L}(X))$ and $\hat{v}(X)=v^{+}_{\delta}(\mathcal{L}(X))$, which are defined because $\mathcal{L}(X)\in\mathcal{D}$ for $X\in\mathcal{D}^{\Lambda}$, and let $-\hat{v}:\mathcal{D}^{\Lambda}\to\mathbb{R}$ be the function whose value at $X$ is the additive inverse of $\hat{v}(X)$. We write $\lVert Z\rVert_{L^{2}}^{2}=\lVert Z\rVert_{L^{2}}\lVert Z\rVert_{L^{2}}$, $|\zeta|^{2}=|\zeta||\zeta|$ and $\tfrac{\alpha}{2}$ for the quotient of $\alpha\in\mathbb{R}$ by $2=1+1$. Then the following hold.

1. (The penalty attains a least value)(i) There is $\mu_{\min}\in\mathcal{D}$ with $\mathcal{E}(\mu_{\min})\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$.

2. (The lift is nonempty and the bounds)(ii) The set $\mathcal{D}^{\Lambda}$ is nonempty, and

$$\hat{u}(X)\le b-\delta e_{0},\qquad -\hat{v}(X)\le -b'-\delta e_{0}\qquad\text{for every }X\in\mathcal{D}^{\Lambda}.$$

3. (Closed superlevel sets)(iii) The functions $\hat{u}$ and $-\hat{v}$ have closed superlevel sets in $L^{2}(\Omega;\mathbb{R}^{d})$.

4. (The doubled difference and its quadratic perturbations)(iv) Let $\alpha,\mu\in\mathbb{R}$ be nonnegative, let $q\in L^{2}(\Omega;\mathbb{R}^{d})\times L^{2}(\Omega;\mathbb{R}^{d})$, and let $\Psi:\mathcal{D}^{\Lambda}\times\mathcal{D}^{\Lambda}\to\mathbb{R}$ be the function with value

$$\Psi(X,Y)=\hat{u}(X)-\hat{v}(Y)-\tfrac{\alpha}{2}\lVert X-Y\rVert_{L^{2}}^{2}-\mu\,\bigl|(X,Y)-q\bigr|^{2}$$

at $(X,Y)$. Then $\Psi(X,Y)\le b-b'-2\delta e_{0}$ for every $(X,Y)\in\mathcal{D}^{\Lambda}\times\mathcal{D}^{\Lambda}$, and $\Psi$ has closed superlevel sets in $L^{2}(\Omega;\mathbb{R}^{d})\times L^{2}(\Omega;\mathbb{R}^{d})$.