Score Bounds at Envelope Test Inequalities from the Absorption Slack

In the setting of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation, let $\rho,\sigma>0$ be the discount rate and the noise intensity and let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ be the Hamiltonian, all as fixed in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §data. Assume that the Hamiltonian $\mathcal{H}$ absorbs shifts at noise level $\sigma$ and is bounded at zero momentum at bounded positions. Lifts $\mathcal{H}_{M}$ are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts; sums, real multiples and the pairing $\langle\cdot,\cdot\rangle_{2}$ of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing; the $L^{2}$ norm and laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; and for real $r>0$, $\kappa_{d}(\Sigma_{d,r})$ is the image of the set $\Sigma_{d,r}$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws under the canonical map $\kappa_{d}$ of Square-Integrable Noncommutative Laws: Standing Notation §laws.

For every real $r>0$ there is a real $\delta_{1}>0$ such that for all reals $m\ge0$ and $b\ge0$ there is a real $C\ge0$ such that the following holds for every tracial W*-probability space $(H,M,\Omega)$, all $L^{2}$ $d$-tuples $X,P,Q$ of $(H,M,\Omega)$ with $\mathrm{law}(X)\in\kappa_{d}(\Sigma_{d,r})$ and $\lVert P\rVert_{2}\le m$, every real $a$ with $|a|\le b$ and every real $\delta$ with $0<\delta\le\delta_{1}$.

1. (Subsolution side)¶ If

$$\rho a+\mathcal{H}_{M}(X,P+\delta Q)+\frac{\sigma^{2}}{2}\Bigl(\langle Q,P\rangle_{2}+\delta\,\lVert Q\rVert_{2}^{2}\Bigr)\le0,$$

then $\lVert Q\rVert_{2}\le C/\delta$.

2. (Supersolution side)¶ If

$$\rho a+\mathcal{H}_{M}(X,P-\delta Q)+\frac{\sigma^{2}}{2}\Bigl(\langle Q,P\rangle_{2}-\delta\,\lVert Q\rVert_{2}^{2}\Bigr)\ge0,$$

then $\lVert Q\rVert_{2}\le C/\delta$.