  We consider fluid flows, governed by the Navier-Stokes equations, subject to
a steady symmetry-breaking bifurcation and forced by a weak noise acting on a
slow time scale. By generalizing the multiple-scale weakly nonlinear expansion
technique employed in the literature for the response of the Duffing
oscillator, we rigorously derive a stochastically forced Stuart-Landau equation
for the dominant symmetry-breaking mode. The probability density function of
the solution, and of the escape time from one attractor to the other, are then
determined by solving the associated Fokker-Planck equation. The validity of
this reduced order model is tested on the flow past a sudden expansion, for a
given Reynolds number and different noise amplitudes. At a very low numerical
cost, the statistics obtained from the amplitude equation accurately reproduce
those of long-time direct numerical simulations.
