Tangent Inequalities for the Wall Energy along Couplings and for the Wall-Confined Free Energy along Optimal Couplings

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 and let $R>0$ be real. $\mathcal{D}_{R}$, $\mathcal{W}_{R}$ and $F^{R}$ are the domain, the wall energy and the wall force of radius $R$; $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, on $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$, is the wall-confined free energy of $(\mathcal{D}_{0},\mathcal{E}_{0})$ with radius $R$, with score domain $\mathcal{D}_{\Xi}$ and score $\Xi$; and the coupling pairing $\mathcal{J}^{1}_{\gamma}$ is that of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing.

1. (Wall energy)¶ Let $\mu\in\mathcal{D}_{R}$ have a square-integrable wall force of radius $R$, let $\nu\in\mathcal{D}_{R}$, and let $\gamma\in\Pi(\mu,\nu)$ be any coupling. Then

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

2. (Free energy)¶ Let $\mu\in\mathcal{D}_{\Xi}$, let $\nu\in\mathcal{D}$, and let $\gamma\in\Pi(\mu,\nu)$ be an optimal coupling. Then

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