Wall-Confined Free Energies with Regular Plan Subgradients

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: 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 $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$, be its wall-confined free energy with radius $R$, with score domain $\mathcal{D}_{\Xi}$ and score $\Xi$. By The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds, $\mathcal{D}\subseteq\Sigma_{d,R}$, and $\mathcal{D}$ is regarded as a subset of the metric space $(\Sigma_{d,R},W_{2})$ of The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points; $\Sigma_{d,R}$ is the set of laws with norm bound $R$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, and $\kappa_{d}$ and $\kappa_{2d}$ are the canonical maps of Square-Integrable Noncommutative Laws: Standing Notation §laws. For $\mu\in\mathcal{D}_{\Xi}$, $\pi^{\Xi}_{\mu}\in\Sigma^{2}_{2d}$ is the score plan of $\mu$. Bounded plans are those of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans, and $J^{-}$ is the plan subjet with slack $0$.

¶The wall-confined free energy $\mathcal{E}$ of $(\mathcal{D}_{0},\mathcal{E}_{0})$ with radius $R$ has regular plan subgradients if the following holds for every function $\chi:\Sigma^{2}_{d}\to\mathbb{R}$, every $\mu\in\mathcal{D}$ and real $r>0$ such that

$$\mathcal{E}(\nu)-\chi\bigl(\kappa_{d}(\nu)\bigr)\ge\mathcal{E}(\mu)-\chi\bigl(\kappa_{d}(\mu)\bigr)\qquad\text{for every }\nu\in\mathcal{D}\text{ with }W_{2}(\nu,\mu)<r,$$

and every bounded plan $\pi$ at $\mu$ with $\kappa_{2d}(\pi)\in J^{-}\chi(\kappa_{d}(\mu))$: one has $\mu\in\mathcal{D}_{\Xi}$ and $\kappa_{2d}(\pi)=\pi^{\Xi}_{\mu}$.