Constant Plan-Jet Viscosity Subsolutions and Supersolutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\rho>0$ be real, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$, let $(\mathrm{E})$ be the discounted stationary Hamilton--Jacobi equation with discount rate $\rho$ and Hamiltonian $\mathcal{H}$, let $c\in\mathbb{R}$, and let $u:\Sigma^{2}_{d}\to\mathbb{R}$ be the constant function with value $c$. Lifts $\mathcal{H}_{M}$ are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, and $0$ denotes the $L^{2}$ $d$-tuple $(0,\dots,0)$ of a tracial W*-probability space.

1. (Constant subsolutions)¶ If $\rho c+\mathcal{H}_{M}(X,0)\le0$ for every tracial W*-probability space $(H,M,\Omega)$ and every $L^{2}$ $d$-tuple $X$ of it, then $u$ is a plan-jet viscosity subsolution of $(\mathrm{E})$.

2. (Constant supersolutions)¶ If $\rho c+\mathcal{H}_{M}(X,0)\ge0$ for every tracial W*-probability space $(H,M,\Omega)$ and every $L^{2}$ $d$-tuple $X$ of it, then $u$ is a plan-jet viscosity supersolution of $(\mathrm{E})$.