Perron's Method for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws: Existence and Well-Posedness

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, be uniformly continuous on bounded sets, and be either Lipschitz in the momentum with linear growth or quadratic with a convex Lipschitz remainder. Let $(\mathrm{E})$ be the discounted stationary Hamilton--Jacobi equation with discount rate $\rho$ and Hamiltonian $\mathcal{H}$. Metrics are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics; subsolutions, supersolutions and solutions are plan-jet viscosity ones; and lifts $\mathcal{H}_{M}$ are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts.

1. (Perron's method)¶ Let $g_{-}:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, upper semicontinuous and a subsolution of $(\mathrm{E})$, and let $g_{+}:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, lower semicontinuous and a supersolution of $(\mathrm{E})$, with $g_{-}\le g_{+}$. Then there is a bounded continuous solution $u$ of $(\mathrm{E})$ with $g_{-}\le u\le g_{+}$.

2. (Well-posedness)¶ Suppose there is a real $K$ with $|\mathcal{H}_{M}(X,0)|\le K$ for every tracial W*-probability space $(H,M,\Omega)$ and every $L^{2}$ $d$-tuple $X$ of it, where $0=(0,\dots,0)$. Then $(\mathrm{E})$ has exactly one bounded continuous solution.