The Supremum of a Bounded Family of Envelope Viscosity Subsolutions is an Envelope Viscosity Subsolution

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, assume that the Hamiltonian $\mathcal{H}$ absorbs shifts at noise level $\sigma$, is lower shift-semicontinuous at noise level $\sigma$ and is bounded at zero momentum at bounded positions, and assume that $\mathcal{E}$ has closed score.

¶There is a real $\delta_{1}>0$ such that the following holds for every real $\delta_{0}$ with $0<\delta_{0}\le\delta_{1}$. Let $b\ge0$ be real, let $\mathcal{F}$ be a nonempty set of functions $v:\Sigma^{2}_{d}\to\mathbb{R}$, each with $|v|\le b$ on $\Sigma^{2}_{d}$ and each an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$, and let $W:\Sigma^{2}_{d}\to\mathbb{R}$, $W(\lambda)=\sup\{v(\lambda):v\in\mathcal{F}\}$, the least upper bound of a nonempty set of reals bounded above by $b$, as recorded in The Real Numbers: Standing Notation and Background §bounds. Then the following hold.

1. (Bound)¶ $|W|\le b$ on $\Sigma^{2}_{d}$.

2. (Subsolution)¶ $W$ is an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$.