The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation

1. (Conventions)¶ The conventions of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation are in force. For a law $\lambda$, $\mathcal{H}_{\lambda}$ is its complex GNS space as in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §gns; the Hamiltonian below is written $\mathcal{H}$ without subscript, and its lifts $\mathcal{H}_{M}$ are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts.

2. (Pairings)¶ The marginal isometries $V^{1}_{\gamma}$, $V^{2}_{\gamma}$, bounded plans, the momentum norm $|\cdot|_{\mathrm{mom}}$, the plan pairing $\mathcal{J}$ and the coupling pairings $\mathcal{J}^{1}_{\gamma}$, $\mathcal{J}^{2}_{\gamma}$ are those of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws. For $\mu\in\Sigma_{d}$ and a bounded plan $\pi$ at $\mu$, $\kappa_{2d}(\pi)$ is a plan at $\kappa_{d}(\mu)$: $\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi)=\kappa_{d}(\pi\circ\sigma_{\mathrm{pr}^{1}})$ by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded, and $\sigma_{\mathrm{pr}^{1}}=\iota^{1}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate.

3. (Parameters)¶ $\rho$, $\sigma$ and $R$ are real numbers with $\rho>0$, $\sigma>0$ and $R>0$: the discount rate, the noise intensity and the wall radius.

4. (Free energy)¶ $(\mathcal{D}_{0},\mathcal{E}_{0})$ is a free entropy penalty; $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$, with $\mathcal{D}_{R}$ the domain of the wall energy of radius $R$, and $\mathcal{E}:\mathcal{D}\to\mathbb{R}$ is the wall-confined free energy of $(\mathcal{D}_{0},\mathcal{E}_{0})$ with radius $R$, and $\mathcal{D}_{\Xi}$ and $\Xi$ are the score domain and the score.

5. (Hamiltonian)¶ $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ is a function, the Hamiltonian.

6. (Background)¶ The following results are in force and may be used without restating them: The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score and Plan Jets of Squared Wasserstein Distances at an Optimal Coupling of Bounded Noncommutative Laws.