Proof of 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

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

By The Controlled von Neumann Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §equation, $(\mathrm{E}^{\mathrm{vN}})$ 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 $\mathcal{H}=\mathcal{H}^{\mathrm{vN}}$, which is a function $\Sigma^{2}_{2d}\to\mathbb{R}$ by The Controlled von Neumann Hamiltonian on Phase-Space Noncommutative Laws §hamiltonian; accordingly, by the convention stated in Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall, the penalised sub- and supersolutions and solutions of $(\mathrm{E}^{\mathrm{vN}})$ are those of $(\mathrm{E})$ with this Hamiltonian.

Claims 1 and 2. Since $f$ is uniformly continuous for the metric of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics, $\mathcal{H}^{\mathrm{vN}}$ satisfies the structure condition at bounded positions by The Controlled von Neumann Hamiltonian at Bounded Positions: Evaluation, the Structure Condition and Uniform Continuity in the Momentum §structure, and it is uniformly continuous in the momentum at bounded positions by The Controlled von Neumann Hamiltonian at Bounded Positions: Evaluation, the Structure Condition and Uniform Continuity in the Momentum §momentum. These are the hypotheses on the Hamiltonian in Comparison and Uniqueness under Wasserstein Semicontinuity for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall, whose remaining data (the setting, the domain $\mathcal{D}$ and semicontinuity on $\mathcal{D}$ in $(\Sigma_{d,R},W_{2})$) are those of the present statement. Hence claim 1 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 §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, both with $\mathcal{H}=\mathcal{H}^{\mathrm{vN}}$.

Claim 3. Under the stated tangent assumption, $(\mathcal{D}_{\chi},-\chi^{*})$ 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 pair $(\mathcal{D}_{0},\mathcal{E}_{0})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy is an arbitrary free entropy penalty, and claims 1 and 2 were proved for an arbitrary one. The objects $\mathcal{D}$, $\mathcal{E}$, $\mathcal{D}_{\Xi}$ and $\Xi$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy are determined by $(\mathcal{D}_{0},\mathcal{E}_{0})$ and $R$, so for $(\mathcal{D}_{0},\mathcal{E}_{0})=(\mathcal{D}_{\chi},-\chi^{*})$ the equation $(\mathrm{E}^{\mathrm{vN}})$, the penalised notions and the domain $\mathcal{D}$ are those attached to this pair, and claims 1 and 2 are exactly the statements for it. In that case $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}=\mathcal{D}_{\chi}\cap\mathcal{D}_{R}$ by The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy.