The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation

1. (Data of the equation)¶ The conventions of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation are in force. In particular $\rho,\sigma,R>0$ are the discount rate, the noise intensity and the wall radius (The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters); $(\mathcal{D}_{0},\mathcal{E}_{0})$ is the free entropy penalty and $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, on $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$, its wall-confined free energy with score domain $\mathcal{D}_{\Xi}$ and score $\Xi$ (The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy); $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ is the Hamiltonian (The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §hamiltonian); bounded plans and the plan pairing $\mathcal{J}$ are those of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §pairings; and $(\mathrm{E})$ is the equation of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws §score-form. $J^{+}$ and $J^{-}$ are the plan superjet and plan subjet with slack $0$.

2. (The domain as a metric space)¶ $\mathcal{D}\subseteq\Sigma_{d,R}$ by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds, and $\mathcal{D}$ is regarded as a subset of 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, and $\mathbb{R}$ carries the metric of The Absolute Value Metric on the Real Line; semicontinuity and continuity of real functions on subsets of $\mathcal{D}$ refer to these metrics.

3. (Scores and shifted plans)¶ For $\mu\in\mathcal{D}_{\Xi}$, with $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$ the tracial W*-probability space of The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, the score $\Xi(\mu)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$ by Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space §score, with $L^{2}$ norm $\lVert\Xi(\mu)\rVert_{2}$ as in Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples. For a bounded plan $\pi$ at $\mu\in\mathcal{D}_{\Xi}$ and real $t$, $\pi\oplus t\,\Xi(\mu)\in\Sigma^{2}_{2d}$ is the shift of $\pi$ by $t\,\Xi(\mu)$.

4. (Penalty envelopes)¶ When $\mathcal{D}$ is nonempty, for a bounded $w:\Sigma^{2}_{d}\to\mathbb{R}$ and real $\delta>0$, $w^{-}_{\delta}$ is its upper penalty envelope and $w^{+}_{\delta}$ its lower penalty envelope at level $\delta$, functions $\mathcal{D}\to\mathbb{R}$.

5. (Background)¶ The results of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §background are in force, and so are Tangent Inequalities for the Wall Energy along Couplings and for the Wall-Confined Free Energy along Optimal Couplings, Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance and Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws; they may be used without restating them.