Proof of Conditional Comparison and Uniqueness for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall, Penalised by Voiculescu's Free Entropy

Each result cited below is universally quantified over the data in its own statement.

Claim 1. By Voiculescu's Non-Microstates Free Entropy of a Noncommutative Law §domain, $\mathcal{D}_{\chi}$ is a subset of $\Sigma_{d}$, and by Voiculescu's Non-Microstates Free Entropy of a Noncommutative Law §entropy, $\chi^{*}(\lambda)$ is a real number for every $\lambda\in\mathcal{D}_{\chi}$; hence $-\chi^{*}$ is a function $\mathcal{D}_{\chi}\to\mathbb{R}$. The pair $(\mathcal{D}_{\chi},-\chi^{*})$ satisfies the lower bound clause by Minus the Free Entropy Is Bounded Below on Bounded Laws and Has Weak-Star Closed Sublevel Sets §lower, the closed sublevel set clause by Minus the Free Entropy Is Bounded Below on Bounded Laws and Has Weak-Star Closed Sublevel Sets §closed, and the tangent inequality clause by the hypothesis of the statement. These are the three properties required in Free Entropy Penalties: Displacement Convexity with Minus the Conjugate Variables as Gradient, so $(\mathcal{D}_{\chi},-\chi^{*})$ is a free entropy penalty.

Claims 2 and 3. By claim 1, the pair $(\mathcal{D}_{0},\mathcal{E}_{0})=(\mathcal{D}_{\chi},-\chi^{*})$ is a free entropy penalty, as The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy requires, so the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation may be adopted with it. Comparison and Uniqueness for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall on Noncommutative Laws is stated in that setting for an arbitrary free entropy penalty; with this penalty its domain $\mathcal{D}$, its data $f$, $(b_{\mu})$, $\mathcal{H}^{\mathrm{LQ}}$, its equation $(\mathrm{E}^{\mathrm{LQ}})$ and its hypotheses on $A$, $c$ and $f$ are exactly those of claims 2 and 3. Hence claim 2 is Comparison and Uniqueness for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall on Noncommutative Laws §comparison and claim 3 is Comparison and Uniqueness for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall on Noncommutative Laws §uniqueness, both applied to the free entropy penalty $(\mathcal{D}_{\chi},-\chi^{*})$.