Proof of Existence and Uniqueness of a Bounded Continuous Plan-Jet Viscosity Solution of the Linear-Quadratic Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws

Each result cited is universally quantified over the data in its own statement. By The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws §equation, $(\mathrm{LQ})$ is the discounted stationary Hamilton--Jacobi equation with discount rate $\rho$ and Hamiltonian $\mathcal{H}^{\mathrm{LQ}}$. By The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure §structure and The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure §quadratic, $\mathcal{H}^{\mathrm{LQ}}$ satisfies the structure condition and is quadratic with a convex Lipschitz remainder; by The Linear-Quadratic Hamiltonian is Uniformly Continuous on Bounded Sets and Bounded at Zero Momentum §uniform it is uniformly continuous on bounded sets. Since $f$ is bounded, there is a real $K$ with $|f|\le K$, and The Linear-Quadratic Hamiltonian is Uniformly Continuous on Bounded Sets and Bounded at Zero Momentum §zero gives $|\mathcal{H}^{\mathrm{LQ}}_{M}(X,0)|\le K$ for every tracial W*-probability space and every $L^{2}$ $d$-tuple $X$. Hence Perron's Method for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws: Existence and Well-Posedness §well-posed applies: $(\mathrm{LQ})$ has exactly one bounded continuous plan-jet viscosity solution.