Envelope Viscosity Solutions versus Free-Energy-Penalised Viscosity Solutions: Envelopes of Semicontinuous Functions, and Semicontinuous Envelope Solutions are Penalised Solutions under Absorption

In the setting of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation, suppose $\mathcal{D}$ is nonempty and let $u:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, with penalty envelopes $u^{\mp}_{\delta}$ as in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §envelopes.

1. (Envelopes of semicontinuous functions)¶ If the function $\mu\mapsto u(\kappa_{d}(\mu))$ on $\mathcal{D}$ is upper semicontinuous on $\mathcal{D}$, then $u^{-}_{\delta}(\mu)=u(\kappa_{d}(\mu))-\delta\,\mathcal{E}(\mu)$ for every real $\delta>0$ and every $\mu\in\mathcal{D}$. ¶If it is lower semicontinuous on $\mathcal{D}$, then $u^{+}_{\delta}(\mu)=u(\kappa_{d}(\mu))+\delta\,\mathcal{E}(\mu)$ for every real $\delta>0$ and every $\mu\in\mathcal{D}$.

2. (Under absorption, semicontinuous envelope solutions are free-energy-penalised solutions) Assume that the Hamiltonian $\mathcal{H}$ absorbs shifts at noise level $\sigma$, and let $\delta_{0}>0$ be real. ¶If $\mu\mapsto u(\kappa_{d}(\mu))$ is upper semicontinuous on $\mathcal{D}$ and $u$ is an envelope viscosity subsolution of $(\mathrm{E})$ with shift range $\delta_{0}$, then $u$ is a free-energy-penalised viscosity subsolution of $(\mathrm{E})$. ¶If $\mu\mapsto u(\kappa_{d}(\mu))$ is lower semicontinuous on $\mathcal{D}$ and $u$ is an envelope viscosity supersolution of $(\mathrm{E})$ with shift range $\delta_{0}$, then $u$ is a free-energy-penalised viscosity supersolution of $(\mathrm{E})$.