Plan-Jet Viscosity Subsolutions, Supersolutions and Solutions of the Discounted 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, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$, consider the discounted stationary Hamilton--Jacobi equation $(\mathrm{E})$ with discount rate $\rho$ and Hamiltonian $\mathcal{H}$, and let $u:\Sigma^{2}_{d}\to\mathbb{R}$. The lifts $\mathcal{H}_{M}$ are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, sums of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing, the $L^{2}$ norm is that of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples, and $J^{+}_{\delta}\varphi(\mu)$ and $J^{-}_{\delta}\varphi(\mu)$ are the plan superjet and the plan subjet of a function $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$ at $\mu\in\Sigma^{2}_{d}$ with slack $\delta\ge0$. Local maxima and local minima of $u-\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, $(u-\varphi)(\nu)=u(\nu)-\varphi(\nu)$, are taken relative to $\Sigma^{2}_{d}$ in the metric space of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics.

1. (Subsolutions)¶ $u$ is a plan-jet viscosity subsolution of $(\mathrm{E})$ if for every real $\delta\ge0$, every $\mu\in\Sigma^{2}_{d}$, every $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$ such that $u-\varphi$ has a local maximum at $\mu$, every $\pi\in J^{+}_{\delta}\varphi(\mu)$ and every real $\eta>0$, there are a tracial W*-probability space $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $X,P,Q$ of $(H,M,\Omega)$ with $\mathrm{law}(X,P)=\pi$, $\lVert Q\rVert_{2}\le\delta$ and

2. (Supersolutions)¶ $u$ is a plan-jet viscosity supersolution of $(\mathrm{E})$ if for every real $\delta\ge0$, every $\mu\in\Sigma^{2}_{d}$, every $\varphi:\Sigma^{2}_{d}\to\mathbb{R}$ such that $u-\varphi$ has a local minimum at $\mu$, every $\pi\in J^{-}_{\delta}\varphi(\mu)$ and every real $\eta>0$, there are a tracial W*-probability space $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $X,P,Q$ of $(H,M,\Omega)$ with $\mathrm{law}(X,P)=\pi$, $\lVert Q\rVert_{2}\le\delta$ and

3. (Solutions)¶ $u$ is a plan-jet viscosity solution of $(\mathrm{E})$ if it is both a plan-jet viscosity subsolution and a plan-jet viscosity supersolution of $(\mathrm{E})$.