Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy on Bounded Noncommutative Laws

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation, let $\mathcal{E}:\mathcal{D}\to\mathbb{R}$ be as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy, and assume $\mathcal{D}\ne\varnothing$. Fix a real $e$ with $e\le\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$, as provided by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds. Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$, let $b\ge0$ be real with $|u(\lambda)|\le b$ and $|v(\lambda)|\le b$ for every $\lambda\in\Sigma^{2}_{d}$, and let $u$ be weak-star upper semicontinuous on bounded laws and $v$ weak-star lower semicontinuous on bounded laws. For reals $\delta\ge0$ and $\alpha>0$ let $\Psi_{\delta,\alpha}:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ be

$$\Psi_{\delta,\alpha}(\mu,\nu)=u\bigl(\kappa_{d}(\mu)\bigr)-v\bigl(\kappa_{d}(\nu)\bigr)-\tfrac{\alpha}{2}W_{2}(\mu,\nu)^{2}-\delta\,\mathcal{E}(\mu)-\delta\,\mathcal{E}(\nu).$$

1. (Bounds)¶ For all reals $\delta\ge0$ and $\alpha>0$, every value of $\Psi_{\delta,\alpha}$ is at most $2b-2\delta e$, and the supremum $S(\delta,\alpha)$ of the values of $\Psi_{\delta,\alpha}$ is a real number; moreover $S(\delta,\alpha)\le S(0,\alpha)-2\delta e$.

2. (Diagonal)¶ For every real $\alpha>0$, $S(0,\alpha)\ge u(\kappa_{d}(\lambda))-v(\kappa_{d}(\lambda))$ for every $\lambda\in\mathcal{D}$.

3. (Monotone in the strength)¶ $S(0,\alpha')\ge S(0,\alpha)$ for all reals $\alpha,\alpha'$ with $0<\alpha'\le\alpha$.

4. (Maximiser)¶ For all reals $\delta>0$ and $\alpha>0$ there is $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ with $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})=S(\delta,\alpha)$, called a maximising pair of $\Psi_{\delta,\alpha}$.

5. (Doubling strength)¶ If $\delta>0$, $0<\alpha'<\alpha$ and $(\hat{\mu},\hat{\nu})$ is a maximising pair of $\Psi_{\delta,\alpha}$, then

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

6. (Vanishing weight)¶ For all reals $\alpha>0$ and $\varepsilon>0$ there is a real $\delta_{1}>0$ with $S(\delta,\alpha)\ge S(0,\alpha)-\varepsilon$ for every real $\delta$ with $0<\delta\le\delta_{1}$.