The Discounted Stationary 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 $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$, the Hamiltonian; the lifts $\mathcal{H}_{M}$, and the lifts $u_{M}$ of a function $u:\Sigma^{2}_{d}\to\mathbb{R}$, are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts.

¶The discounted stationary Hamilton--Jacobi equation on $\Sigma^{2}_{d}$ with discount rate $\rho$ and Hamiltonian $\mathcal{H}$, for an unknown function $u:\Sigma^{2}_{d}\to\mathbb{R}$, is written formally as

for every tracial W*-probability space $(H,M,\Omega)$ and every $L^{2}$ $d$-tuple $X$ of $(H,M,\Omega)$, where $\nabla u_{M}(X)$ is a formal symbol for the gradient of $u_{M}$ at $X$.