Comparison Principle for Plan-Jet Viscosity Solutions 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 and let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ satisfy the structure condition. Let $(\mathrm{E})$ be the discounted stationary Hamilton--Jacobi equation with discount rate $\rho$ and Hamiltonian $\mathcal{H}$. Let $u,v:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, let $u$ be upper semicontinuous and a plan-jet viscosity subsolution of $(\mathrm{E})$, and let $v$ be lower semicontinuous and a plan-jet viscosity supersolution of $(\mathrm{E})$; metrics are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics.

1. (Hamiltonians Lipschitz in the momentum)¶ If $\mathcal{H}$ is Lipschitz in the momentum with linear growth, then $u(\mu)\le v(\mu)$ for every $\mu\in\Sigma^{2}_{d}$.

2. (Quadratic Hamiltonians)¶ If $\mathcal{H}$ is quadratic with a convex Lipschitz remainder, then $u(\mu)\le v(\mu)$ for every $\mu\in\Sigma^{2}_{d}$.