The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise

In the setting of Unitary Laws with Free Unitary Noise: Standing Data, with the conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces, an arbitrary map $H:\mathcal{L}_{d}\times E_{d}\to\mathbb{R}$, called the Hamiltonian of the equation, the constants $\rho,\beta$, the source $f$ and the noise direction $\widehat{\Theta}\lambda\in E_{d}$, the Hamilton–Jacobi–Bellman equation on unitary laws with Hamiltonian $H$ and free unitary noise for an unknown $V:\mathcal{L}_{d}\to\mathbb{R}$ is the relation¶

$$\rho\,V(\lambda)+H(\lambda,p)-\frac{\beta^{2}}{2}\,\bigl\langle p,\widehat{\Theta}\lambda\bigr\rangle_{d}=f(\lambda),\qquad\lambda\in\mathcal{L}_{d},$$

between the value $V(\lambda)$ and a momentum $p\in E_{d}$, where $p$ stands in for a gradient of $V$ at $\lambda$ in the gauge space $E_{d}$ and $\langle\cdot,\cdot\rangle_{d}$ is the gauge inner product. The equation is given a meaning through test functions $F\in C^{1}(E_{d})$ as in Unitary Laws with Free Unitary Noise: Standing Data §test-functions, with $p=DF(\iota_{d}(\lambda))$, in the sense of viscosity solutions.