 Solution

For the constitutive system, every propensity is affine in the copy numbers. The exact first- and second-moment equations therefore close: expectations of drift are determined by the means, and covariance drift by the covariance. The stationary fluctuation-dissipation calculation is consequently exact, even though the full joint stationary distribution need not be Gaussian.

For feedback, expectations such as $E[x_{1}f(x_2)]$ involve nonlinear moments and correlations. Replacing them by a linear expansion about the mean is a linear noise approximation, so the covariance formula is approximate. It works well around a stable single equilibrium when relative fluctuations are small and the rates vary smoothly over their typical range, generally at sufficiently large molecule numbers. Low counts, strong nonlinearities, multiple attractors and proximity to instability can invalidate it.