  We adopt an input-output approach to analyze the effect of persistent
white-in-time 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. 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.
Our approach does not rely on costly stochastic simulations or adjoint-based
sensitivity analyses. We use our framework to study the destabilizing effects
of stochastic base flow variations in transitional parallel flows, the
reliability of numerically estimated mean velocity profiles in turbulent
channel flows, and the robust performance of a typical boundary control
strategy for turbulence suppression in the wake of parametric uncertainties. We
explain the robust amplification of streamwise streaks in the presence of
streamwise base flow variations and explore the possibility of modeling
statistical signatures of turbulent flows by shaping the statistics of
multiplicative sources of excitation in the linearized dynamics.
