Comparison and Uniqueness under Wasserstein Semicontinuity for the Controlled von Neumann HJB Equation with Free Langevin Noise in a Wall, Including the Free Entropy Penalty

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation, let $f$ and $\mathcal{H}^{\mathrm{vN}}$ be as in The Controlled von Neumann Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §data, let $(\mathrm{E}^{\mathrm{vN}})$ be the equation of The Controlled von Neumann Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §equation, which is the equation of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §score-form with the Hamiltonian $\mathcal{H}=\mathcal{H}^{\mathrm{vN}}$, let $R$ be the wall radius of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters, and let $\mathcal{D}$ be the domain of the wall-confined free energy as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy. Semicontinuity on $\mathcal{D}$ refers to 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, which contains $\mathcal{D}$ by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds. Assume that $f$ is uniformly continuous on $\Sigma^{2}_{d}$ with the metric of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics.

1. (Comparison)¶ Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded; let $u$ be a free-energy-penalised viscosity subsolution of $(\mathrm{E}^{\mathrm{vN}})$ such that the function $\mu\mapsto u(\kappa_{d}(\mu))$ on $\mathcal{D}$ is upper semicontinuous on $\mathcal{D}$, and let $v$ be a free-energy-penalised viscosity supersolution of $(\mathrm{E}^{\mathrm{vN}})$ such that the function $\mu\mapsto v(\kappa_{d}(\mu))$ on $\mathcal{D}$ is lower semicontinuous on $\mathcal{D}$. Then $u(\kappa_{d}(\mu))\le v(\kappa_{d}(\mu))$ for every $\mu\in\mathcal{D}$.

2. (Uniqueness)¶ Any two bounded free-energy-penalised viscosity solutions $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ of $(\mathrm{E}^{\mathrm{vN}})$ such that the functions $\mu\mapsto u(\kappa_{d}(\mu))$ and $\mu\mapsto v(\kappa_{d}(\mu))$ on $\mathcal{D}$ are both upper semicontinuous and lower semicontinuous on $\mathcal{D}$ agree at $\kappa_{d}(\mu)$ for every $\mu\in\mathcal{D}$.

3. (Free entropy penalty)¶ Let $\mathcal{D}_{\chi}$ and $\chi^{*}$ be the domain and the free entropy of Free Products, Semicircular Laws, the Free Heat Flow, Free Fisher Information and Free Entropy: Standing Notation §entropy, let $-\chi^{*}:\mathcal{D}_{\chi}\to\mathbb{R}$ be the map $\lambda\mapsto-\chi^{*}(\lambda)$, and assume that the pair $(\mathcal{D}_{\chi},-\chi^{*})$ satisfies the tangent inequality clause of a free entropy penalty, as conjectured in Displacement Convexity of Minus the Free Entropy: the Tangent Inequality along Optimal Couplings §tangent. Then $(\mathcal{D}_{\chi},-\chi^{*})$ is a free entropy penalty by Conditional Comparison and Uniqueness for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall, Penalised by Voiculescu's Free Entropy §penalty, and consequently claims 1 and 2 apply to $(\mathcal{D}_{0},\mathcal{E}_{0})=(\mathcal{D}_{\chi},-\chi^{*})$, for which $\mathcal{D}=\mathcal{D}_{\chi}\cap\mathcal{D}_{R}$.