Absorption of Shifts by a Hamiltonian on Phase-Space Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$, with lifts $\mathcal{H}_{M}$ as in Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, and let $\sigma>0$ be real. Sums and real multiples of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing; the $L^{2}$ norm $\lVert\cdot\rVert_{2}$ and laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; $\Sigma_{d,r}$ is the set of laws with norm bound $r$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws and $\kappa_{d}$ the canonical map of Square-Integrable Noncommutative Laws: Standing Notation §laws.

¶$\mathcal{H}$ absorbs shifts at noise level $\sigma$ if for every real $r>0$ there are a real $\delta_{0}>0$ and a nondecreasing function $\omega:[0,\infty)\to[0,\infty)$ such that, 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 every real $\delta$ with $0<\delta\le\delta_{0}$,

$$\bigl|\mathcal{H}_{M}(X,P+\delta Q)-\mathcal{H}_{M}(X,P)\bigr|\le\delta\,\omega\bigl(\lVert P\rVert_{2}\bigr)+\frac{\sigma^{2}\delta}{4}\,\lVert Q\rVert_{2}^{2}.$$