The Wall-Confined Free Energy and Its 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 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 $R$, and $\xi_{\lambda}=(\xi_{\lambda,1},\dots,\xi_{\lambda,d})$ denotes the conjugate variables of a law $\lambda$ that has them.

1. (Energy)¶ The wall-confined free energy of $(\mathcal{D}_{0},\mathcal{E}_{0})$ with radius $R$ is the function $\mathcal{E}:\mathcal{D}\to\mathbb{R}$ on $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$ given by $\mathcal{E}(\lambda)=\mathcal{E}_{0}(\lambda)+\mathcal{W}_{R}(\lambda)$.

2. (Score)¶ The score domain $\mathcal{D}_{\Xi}$ is the set of $\lambda\in\mathcal{D}$ that have conjugate variables and a square-integrable wall force of radius $R$. For $\lambda\in\mathcal{D}_{\Xi}$ the score of $\lambda$ is

$$\Xi(\lambda)=\bigl(F^{R}_{1}(\lambda)-\xi_{\lambda,1},\dots,F^{R}_{d}(\lambda)-\xi_{\lambda,d}\bigr)\in\mathcal{H}_{\lambda}^{d}.$$