The Controlled von Neumann Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws

We work in the setting of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation, with $\rho$ and $\sigma$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters, $\mathcal{D}_{\Xi}$ and $\Xi$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy, bounded plans and the plan pairing $\mathcal{J}$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §pairings, and $J^{+}$, $J^{-}$ the plan superjet and plan subjet with slack $0$.

1. (Data)¶ Let $f:\Sigma^{2}_{d}\to\mathbb{R}$ and let $\mathcal{H}^{\mathrm{vN}}:\Sigma^{2}_{2d}\to\mathbb{R}$ be the controlled von Neumann Hamiltonian with running cost $f$.

2. (Equation)¶ The controlled von Neumann Hamilton-Jacobi-Bellman equation with free Langevin noise in a wall 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 $\mathcal{H}=\mathcal{H}^{\mathrm{vN}}$: for a function $U:\Sigma^{2}_{d}\to\mathbb{R}$, a law $\mu\in\mathcal{D}_{\Xi}$ and a bounded plan $\pi$ at $\mu$ with $\kappa_{2d}(\pi)\in J^{+}U(\kappa_{d}(\mu))\cap J^{-}U(\kappa_{d}(\mu))$,

$$(\mathrm{E}^{\mathrm{vN}})\qquad\rho\,U\bigl(\kappa_{d}(\mu)\bigr)+\mathcal{H}^{\mathrm{vN}}\bigl(\kappa_{2d}(\pi)\bigr)+\frac{\sigma^{2}}{2}\,\mathcal{J}\bigl(\Xi(\mu),\pi\bigr)=0.$$