Exact Finite Response of Higher-Order Statistics in Nonequilibrium Networks

We develop an exact finite-perturbation response theory for the higher-order statistics of general state observables in nonequilibrium Markov networks. For local perturbations, the moment generating function obeys an exact nonlinear response identity in the perturbation strength, controlled by a single kinetic parameter α. This induces an exact nonlinear map for the cumulant generating function and closed expressions for finite responses of all cumulants through partial Bell polynomials. As a central application, we obtain a complete theory of variance control, including sharp saturation limits, a classification of monotone-suppression and sign-reversal regimes, and meanfirst-passage-time design principles for optimal noise suppression. The same framework yields exact formulas for operational and symmetrized response resolution along the accessible control branch. A four-state gene-regulation model illustrates how local kinetic control suppresses the occupancy noise of a transcriptionally competent promoter state.