The Bump Construction for Envelope Viscosity Subsolutions

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 upper shift-semicontinuous at noise level $\sigma$ and is bounded at zero momentum at bounded positions, and assume that $\mathcal{E}$ has closed score and has regular plan subgradients.

¶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 $g:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded and an envelope viscosity supersolution of $(\mathrm{E})$ with shift range $\delta_{0}$, and let $w:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded and an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$, with $w\le g$ on $\Sigma^{2}_{d}$. If $w$ is not an envelope viscosity supersolution of $(\mathrm{E})$ with shift range $\delta_{0}$, then there is a bounded $U:\Sigma^{2}_{d}\to\mathbb{R}$ that is an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$, with $w\le U\le g$ on $\Sigma^{2}_{d}$ and $U(\kappa_{d}(\nu))>w(\kappa_{d}(\nu))$ for some $\nu\in\mathcal{D}$.