Comparison and Uniqueness 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 $\mathcal{D}$ 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, 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.

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