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

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\rho>0$, $f$ and the affine data $b_{\mu}$ satisfy the hypotheses of The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure, and let $(\mathrm{LQ})$ be the linear-quadratic Hamilton--Jacobi equation with discount rate $\rho$, drift $b$ and running cost $f$. Metrics are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics.

¶$(\mathrm{LQ})$ has exactly one bounded continuous plan-jet viscosity solution $u:\Sigma^{2}_{d}\to\mathbb{R}$.