The Wall Energy: Finiteness Forces the Norm Bound, Closed Sublevel Sets, and Monotonicity of the Wall Force along Couplings

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $R>0$ be real, and let $\mathcal{D}_{R}$, $\mathcal{W}_{R}$ and $F^{R}$ be the domain, the wall energy and the wall force of radius $R$. 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. (Norm bound)¶ $\mathcal{D}_{R}\subseteq\Sigma_{d,R}$, and $\mathcal{W}_{R}(\lambda)\ge0$ for every $\lambda\in\mathcal{D}_{R}$.

2. (Closed sublevel sets)¶ Let $c$ be real and let $(\lambda_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathcal{D}_{R}$ with $\mathcal{W}_{R}(\lambda_{m})\le c$ for every $m\in\mathbb{N}$ that converges weak-star to some $\lambda\in\Sigma_{d}$. Then $\lambda\in\mathcal{D}_{R}$ and $\mathcal{W}_{R}(\lambda)\le c$.

3. (Monotone force)¶ Let $\mu,\nu\in\Sigma_{d}$ have square-integrable wall forces of radius $R$ and let $\gamma\in\Pi(\mu,\nu)$ be any coupling. Then

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