Proof of Constant Plan-Jet Viscosity Subsolutions and Supersolutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws

Each result cited is universally quantified over the data in its own statement.

Claim 1. We verify the jet form of Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §sub. Let $\delta'\ge0$, $\mu'\in\Sigma^{2}_{d}$, $\pi\in J^{+}_{\delta'}u(\mu')$ and $\eta>0$. By Plan Jets at a Common Realisation: Superdifferential and Subdifferential Momenta are Close, and the Plan Jets of a Constant §constants, $\widehat{M}(\mathrm{pr}^{2}_{\#}\pi)\le\delta'^{2}$. By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling there are a tracial W*-probability space $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $X,P$ of it with $\mathrm{law}(X,P)=\pi$; then $\lVert P\rVert_{2}^{2}=\widehat{M}(\mathrm{pr}^{2}_{\#}\pi)\le\delta'^{2}$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments, so $\lVert P\rVert_{2}\le\delta'$ (Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). Put $Q=(-1)P$, an $L^{2}$ $d$-tuple by Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations, with $\lVert Q\rVert_{2}=\lVert P\rVert_{2}\le\delta'$ and $P+Q=0$. Then, by hypothesis,

Hence $u$ is a plan-jet viscosity subsolution.

Claim 2. Identical, with Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions §super, plans $\pi\in J^{-}_{\delta'}u(\mu')$ (to which Plan Jets at a Common Realisation: Superdifferential and Subdifferential Momenta are Close, and the Plan Jets of a Constant §constants applies equally), and the conclusion $\rho c+\mathcal{H}_{M}(X,0)\ge0>-\eta$.