Constant Barriers for Envelope Viscosity Solutions under Regular Plan Subgradients

In the setting of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation, let $\rho,\sigma,R>0$ be the discount rate, the noise intensity and the wall radius; let $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, on $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$, be the wall-confined free energy with radius $R$ of the free entropy penalty $(\mathcal{D}_{0},\mathcal{E}_{0})$, with score domain $\mathcal{D}_{\Xi}$ and score $\Xi$; let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ be the Hamiltonian; and let $(\mathrm{E})$ be the discounted Hamilton--Jacobi--Bellman equation with free Langevin noise in a wall, in score form, with these data, all as fixed in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §data. Suppose $\mathcal{D}$ is nonempty and $\mathcal{E}$ has regular plan subgradients. Lifts $\mathcal{H}_{M}$ are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, laws of $L^{2}$ $d$-tuples are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples, and $\kappa_{d}(\Sigma_{d,R})$ is the image of the set $\Sigma_{d,R}$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws under the canonical map $\kappa_{d}$ of Square-Integrable Noncommutative Laws: Standing Notation §laws. Let $K\ge0$ be real with $|\mathcal{H}_{M}(X,0)|\le K$ for every tracial W*-probability space $(H,M,\Omega)$ and every $L^{2}$ $d$-tuple $X$ of it with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,R})$, where $0$ is the zero $L^{2}$ $d$-tuple, and let $\delta_{0}>0$ be real.

1. (Lower barrier)¶ The constant function $\Sigma^{2}_{d}\to\mathbb{R}$ with value $-K/\rho$ is an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$.

2. (Upper barrier)¶ The constant function $\Sigma^{2}_{d}\to\mathbb{R}$ with value $K/\rho$ is an envelope viscosity supersolution of $(\mathrm{E})$ with shift range $\delta_{0}$.