The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $(\mathcal{D}_{0},\mathcal{E}_{0})$ be a free entropy penalty, let $R>0$ be real, and let $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, on $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$ with $\mathcal{D}_{R}$ the domain of the wall energy, be its wall-confined free energy with radius $R$, with score domain $\mathcal{D}_{\Xi}$ and score $\Xi$. The coupling pairings $\mathcal{J}^{1}_{\gamma}$ and $\mathcal{J}^{2}_{\gamma}$ are those of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing, and weak-star convergence is that of Weak-Star Convergence of Noncommutative Laws §weak-star.

1. (Bounds)¶ $\mathcal{D}\subseteq\Sigma_{d,R}$, and there is a real $e$ with $e\le\mathcal{E}(\lambda)$ for every $\lambda\in\mathcal{D}$.

2. (Compact sublevel sets)¶ Let $c$ be real and let $(\lambda_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{D}$ with $\mathcal{E}(\lambda_{m})\le c$ for every $m\in\mathbb{N}$. Then there are a strictly increasing sequence $(m_{i})_{i\in\mathbb{N}}$ of natural numbers and $\lambda\in\mathcal{D}$ such that $(\lambda_{m_{i}})_{i\in\mathbb{N}}$ converges weak-star to $\lambda$, the real sequence $(\mathcal{E}(\lambda_{m_{i}}))_{i\in\mathbb{N}}$ converges, and $\mathcal{E}(\lambda)\le\lim_{i\to\infty}\mathcal{E}(\lambda_{m_{i}})$.

3. (Monotone score)¶ If $\mu,\nu\in\mathcal{D}_{\Xi}$ and $\gamma\in\Pi(\mu,\nu)$ is an optimal coupling, then

$$\mathcal{J}^{1}_{\gamma}\bigl(\Xi(\mu)\bigr)-\mathcal{J}^{2}_{\gamma}\bigl(\Xi(\nu)\bigr)\ge0.$$