Proof of Comparison and Uniqueness under Wasserstein Semicontinuity for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall, Including the Free Entropy Penalty

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

Step 1 (The Hamiltonian). By The Linear-Quadratic Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §data, $\mathcal{H}^{\mathrm{LQ}}$ is the linear-quadratic Hamiltonian built from $f$ and $(b_{\mu})$, which is the Hamiltonian of The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure and The Linear-Quadratic Hamiltonian is Uniformly Continuous on Bounded Sets and Bounded at Zero Momentum; the reals $a,L\ge0$ and the assumptions on $f$ stated here are exactly the hypotheses of those lemmas. By The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure §structure, $\mathcal{H}^{\mathrm{LQ}}$ satisfies the structure condition, hence the structure condition at bounded positions by The Structure Condition and Uniform Continuity on Bounded Sets Imply Their Forms at Bounded Positions §structure. By The Linear-Quadratic Hamiltonian is Uniformly Continuous on Bounded Sets and Bounded at Zero Momentum §uniform, $\mathcal{H}^{\mathrm{LQ}}$ is uniformly continuous on bounded sets, hence uniformly continuous in the momentum at bounded positions by The Structure Condition and Uniform Continuity on Bounded Sets Imply Their Forms at Bounded Positions §momentum.

Step 2 (Claims 1 and 2). By The Linear-Quadratic Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §equation, $(\mathrm{E}^{\mathrm{LQ}})$ is the equation $(\mathrm{E})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §score-form with the Hamiltonian of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §hamiltonian taken to be $\mathcal{H}^{\mathrm{LQ}}:\Sigma^{2}_{2d}\to\mathbb{R}$, and the penalised sub- and supersolutions of $(\mathrm{E}^{\mathrm{LQ}})$ are those of Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall for this Hamiltonian. By Step 1 this Hamiltonian satisfies both hypotheses of Comparison and Uniqueness under Wasserstein Semicontinuity for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall. Claim 1 is therefore Comparison and Uniqueness under Wasserstein Semicontinuity for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §comparison, and claim 2 is Comparison and Uniqueness under Wasserstein Semicontinuity for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall §uniqueness, applied with $\mathcal{H}=\mathcal{H}^{\mathrm{LQ}}$.

Step 3 (Claim 3). Under the stated tangent hypothesis, $(\mathcal{D}_{\chi},-\chi^{*})$ 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. The setting The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation admits any free entropy penalty as $(\mathcal{D}_{0},\mathcal{E}_{0})$ (The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy), with $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$; and Steps 1 and 2 use no property of $(\mathcal{D}_{0},\mathcal{E}_{0})$ beyond its being a free entropy penalty, the data $f$, $(b_{\mu})$, $a$, $L$ and $R$ being unchanged. Hence claims 1 and 2 hold for $(\mathcal{D}_{0},\mathcal{E}_{0})=(\mathcal{D}_{\chi},-\chi^{*})$, with $\mathcal{D}=\mathcal{D}_{\chi}\cap\mathcal{D}_{R}$.