Free Entropy Penalties: Displacement Convexity with Minus the Conjugate Variables as Gradient

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, the conjugate variables of a law $\lambda$ that has them are written $\xi_{\lambda}$, the coupling pairing $\mathcal{J}^{1}_{\gamma}$ is that 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.

A free entropy penalty is a pair $(\mathcal{D}_{0},\mathcal{E}_{0})$ of a set $\mathcal{D}_{0}\subseteq\Sigma_{d}$ and a function $\mathcal{E}_{0}:\mathcal{D}_{0}\to\mathbb{R}$ with the following three properties.

1. (Lower bound)¶ For every real $R>0$ there is a real $e$ with $e\le\mathcal{E}_{0}(\lambda)$ for every $\lambda\in\mathcal{D}_{0}\cap\Sigma_{d,R}$.

2. (Closed sublevel sets)¶ For all reals $R>0$ and $c$, and every sequence $(\lambda_{m})_{m\in\mathbb{N}}$ in $\mathcal{D}_{0}\cap\Sigma_{d,R}$ with $\mathcal{E}_{0}(\lambda_{m})\le c$ for every $m\in\mathbb{N}$ that converges weak-star to some $\lambda\in\Sigma_{d}$, one has $\lambda\in\mathcal{D}_{0}$ and $\mathcal{E}_{0}(\lambda)\le c$.

3. (Tangent inequality)¶ For every $\mu\in\mathcal{D}_{0}$ that has conjugate variables, every $\nu\in\mathcal{D}_{0}$ and every optimal coupling $\gamma\in\Pi(\mu,\nu)$,

$$\mathcal{E}_{0}(\nu)\ge\mathcal{E}_{0}(\mu)+\mathcal{J}^{1}_{\gamma}(\xi_{\mu}).$$