The Discounted 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, the score domain $\mathcal{D}_{\Xi}$ and the score $\Xi$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy, the Hamiltonian $\mathcal{H}$ as in The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §hamiltonian, and 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. $J^{+}$ and $J^{-}$ are the plan superjet and the plan subjet with slack $0$.

¶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 $\kappa_{2d}(\pi)\in J^{+}U(\kappa_{d}(\mu))\cap J^{-}U(\kappa_{d}(\mu))$, the discounted Hamilton-Jacobi-Bellman equation with free Langevin noise in a wall, in score form, is

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