Establish a global curvature certificate for the deflated value

Prove that the deleted-noise curvature condition for the deflated Feynman–Kac value holds throughout the relevant state–time domain of the nonlinear stochastic swing-network application, thereby establishing the deflated value as a global lower bound on the optimal control cost.

Background

The noise-deflation construction produces a lower bound only when the directional curvature condition tr⁡(D~λ∇2Jλ)≥0\operatorname{tr}(\widetilde D_\lambda\nabla^2 J_\lambda)\ge 0 holds throughout the domain. In the nonlinear swing-network application, the paper verifies positivity only on a terminal layer and at finitely many sampled states, so the resulting deflated values are not globally certified lower bounds.

The authors explain that generic derivative bounds are too conservative to yield a domain-wide enclosure. A successful resolution would require a sharper structural argument, such as a semigroup or comparison principle, a monotonicity result, or a covariance-sensitive estimate based on the Gibbs representation.

References

The distinction that matters most is between the completed value $+$, which is an exact lower bound whose numerical value is a Monte Carlo estimate, and the deflated value $$, which is a lower bound only if a curvature condition holds that we have not verified on the whole domain.

— How suboptimal is my stochastic network controller allowed to be? Completion certificates with application to power grids hosting AI data centers  (2610.03275 - Chertkov, 2 Oct 2026) in Section 7, Discussion, subsection “What is certified, and how”; Appendix, Section “Verifying the curvature condition for the deflated value,” subsection “Why a domain-wide enclosure is out of reach”

It is the appropriate framework for a certified synchrony-loss probability and should not be confused with the smooth performance value used in the main text; we have not constructed such a $Z$ for the event studied here, and the crossing probabilities of \Cref{sec:numerics} are empirical.

— How suboptimal is my stochastic network controller allowed to be? Completion certificates with application to power grids hosting AI data centers  (2610.03275 - Chertkov, 2 Oct 2026) in Appendix, Section “First-exit probability as a separate certificate”; Section 7, Discussion, subsection “Limitations”