We derive simplified formulas for analyzing the stability of stochastic
parametrically forced linear systems. This extends the results in [T. Blass and
L.A. Romero, SIAM J. Control Optim. 51(2):1099--1127, 2013] where, assuming the
stochastic excitation is small, the stability of such systems was computed
using a weighted sum of the extended power spectral density over the
eigenvalues of the unperturbed operator. In this paper, we show how to convert
this to a sum over the residues of the extended power spectral density. For
systems where the parametric forcing term is a rank one matrix, this leads to
an enormous simplification.Comment: 16 page