Comparison and Uniqueness under Wasserstein Semicontinuity for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall

In the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation, consider the equation $(\mathrm{E})$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §score-form, 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 $\mathcal{D}$ 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, and assume that the Hamiltonian $\mathcal{H}$ of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §hamiltonian satisfies the structure condition at bounded positions and is uniformly continuous in the momentum at bounded positions. 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.

1. (Comparison)¶ Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded; let $u$ be a free-energy-penalised viscosity subsolution of $(\mathrm{E})$ 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 $(\mathrm{E})$ 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)¶ Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded free-energy-penalised viscosity solutions of $(\mathrm{E})$ 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 $\mathcal{D}$. Then $u(\kappa_{d}(\mu))=v(\kappa_{d}(\mu))$ for every $\mu\in\mathcal{D}$.