  The classical fluctuation-dissipation theorem predicts the average response
of a dynamical system to an external deterministic perturbation via time-lagged
statistical correlation functions of the corresponding unperturbed system. In
this work we develop a fluctuation-response theory and test a computational
framework for the leading order response of statistical averages of a
deterministic or stochastic dynamical system to an external stochastic
perturbation. In the case of a stochastic unperturbed dynamical system, we
compute the leading order fluctuation-response formulas for two different
cases: when the existing stochastic term is perturbed, and when a new,
statistically independent, stochastic perturbation is introduced. We
numerically investigate the effectiveness of the new response formulas for an
appropriately rescaled Lorenz 96 system, in both the deterministic and
stochastic unperturbed dynamical regimes.
