Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $u,\varphi:\Sigma^{2}_{d}\to\mathbb{R}$, let $\mu\in\Sigma^{2}_{d}$ and let $\delta\ge0$ be real. The plan superjets $J^{+}_{\delta}$ and plan subjets $J^{-}_{\delta}$ are those of Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §subjet; local maxima and local minima of $u-\varphi$ are taken relative to $\Sigma^{2}_{d}$ in the metric space of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics; 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 and the $L^{2}$ norm is that of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples.

1. (Touching from above)¶ If $u-\varphi$ has a local maximum at $\mu$, then $J^{+}_{\delta}\varphi(\mu)\subseteq J^{+}_{\delta}u(\mu)$.

2. (Touching from below)¶ If $u-\varphi$ has a local minimum at $\mu$, then $J^{-}_{\delta}\varphi(\mu)\subseteq J^{-}_{\delta}u(\mu)$.

Let moreover $\rho>0$ be real, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$, and let $(\mathrm{E})$ be the discounted stationary Hamilton--Jacobi equation with discount rate $\rho$ and Hamiltonian $\mathcal{H}$.

3. (Jet form of subsolutions)¶ $u$ is a plan-jet viscosity subsolution of $(\mathrm{E})$ if and only if for every real $\delta'\ge0$, every $\mu'\in\Sigma^{2}_{d}$, every $\pi\in J^{+}_{\delta'}u(\mu')$ 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 $\rho\,u(\mu')+\mathcal{H}_{M}(X,P+Q)\le\eta$.

4. (Jet form of supersolutions)¶ $u$ is a plan-jet viscosity supersolution of $(\mathrm{E})$ if and only if for every real $\delta'\ge0$, every $\mu'\in\Sigma^{2}_{d}$, every $\pi\in J^{-}_{\delta'}u(\mu')$ 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 $\rho\,u(\mu')+\mathcal{H}_{M}(X,P+Q)\ge-\eta$.