Existence, Penalty Bounds and Monotonicity of the Maximum of the Wasserstein-Doubled Difference of Bounded Functions

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 $\mathcal{P}_{2}(\mathbb{R}^{d})$, and fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$, as provided by Basic Properties of a Wasserstein-Coercive Penalty Pair (ii). Let $u,v:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\mu)\le b$ and $b'\le v(\mu)$ for every $\mu\in\mathcal{D}$. 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, so that for positive $\delta\in\mathbb{R}$ the $\delta$-envelopes $u^{-}_{\delta}$ and $v^{+}_{\delta}$, functions on $\mathcal{D}$, are defined. For positive $\delta,\alpha\in\mathbb{R}$ let $\Psi_{\delta,\alpha}:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ have the value

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

where $\tfrac{\alpha}{2}$ is the product of $\alpha$ with the multiplicative inverse of $2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field). Then the following hold for all positive $\delta,\alpha\in\mathbb{R}$.

1. (A maximiser exists)(i) Every value of $\Psi_{\delta,\alpha}$ is at most $b-b'-2\delta e_{0}$; the supremum $M(\delta,\alpha)$ of these values is a real number; and there is $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ with $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})=M(\delta,\alpha)$, called a maximising pair of $\Psi_{\delta,\alpha}$.

2. (The penalty where the doubled difference is nonnegative)(ii) If $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ satisfies $0\le\Psi_{\delta,\alpha}(\mu,\nu)$, then

$$\delta\,|\mathcal{E}(\mu)|\le|b|+|b'|+\delta\,|e_{0}|\qquad\text{and}\qquad\delta\,|\mathcal{E}(\nu)|\le|b|+|b'|+\delta\,|e_{0}|,$$

where $|s|$ is the absolute value of $s\in\mathbb{R}$.

3. (Decreasing the penalty weight)(iii) If $\delta'\in\mathbb{R}$ satisfies $0<\delta'<\delta$ and $(\hat{\mu},\hat{\nu})$ is a maximising pair of $\Psi_{\delta,\alpha}$, then

$$M(\delta,\alpha)+(\delta-\delta')\bigl(\mathcal{E}(\hat{\mu})+\mathcal{E}(\hat{\nu})\bigr)\le M(\delta',\alpha).$$

4. (Decreasing the doubling strength)(iv) If $\alpha'\in\mathbb{R}$ satisfies $0<\alpha'<\alpha$ and $(\hat{\mu},\hat{\nu})$ is a maximising pair of $\Psi_{\delta,\alpha}$, then

$$M(\delta,\alpha)+\tfrac{\alpha-\alpha'}{2}\,W_{2}(\hat{\mu},\hat{\nu})^{2}\le M(\delta,\alpha').$$