  We adopt an input-output approach to analyze the effect of persistent
white-in-time structured stochastic base flow perturbations on the mean-square
properties of the linearized Navier-Stokes equations. Such base flow variations
enter the linearized dynamics as multiplicative sources of uncertainty that can
alter the stability of the linearized dynamics and their receptivity to
exogenous excitations. Our approach does not rely on costly stochastic
simulations or adjoint-based sensitivity analysis. We provide verifiable
conditions for mean-square stability and study the frequency response of the
flow subject to additive and multiplicative sources of uncertainty using the
solution to the generalized Lyapunov equation. For small-amplitude base flow
perturbations, we bypass the need to solve large generalized Lyapunov equations
by adopting a perturbation analysis. We use our framework to study the
destabilizing effects of stochastic base flow variations in transitional
parallel flows, and the reliability of numerically estimated mean velocity
profiles in turbulent channel flows. We uncover the Reynolds number scaling of
critically destabilizing perturbation variances and demonstrate how the
wall-normal shape of base flow modulations can influence the amplification of
various length scales. Furthermore, we explain the robust amplification of
streamwise streaks in the presence of streamwise base flow variations by
analyzing the dynamical structure of the governing equations as well as the
Reynolds number dependence of the energy spectrum.
