  This article is concerned with the time evolution of the oblique
laminar-turbulent bands of transitional plane Couette flow under the influence
of turbulent noise. Our study is focused on the amplitude of modulation of
turbulence. In order to guide the numerical study of the flow, we first perform
an analytical and numerical analysis of a Stochastic Ginzburg-Landau equation
for a complex order parameter. The modulus of this order parameter models the
amplitude of modulation of turbulence. Firstly, we compute the autocorrelation
function of said modulus once the band is established. Secondly, we perform a
calculation of average and fluctuations around the exponential growth of the
order parameter. This type of analysis is similar to the Stochastic Structural
Stability Theory. We then perform numerical simulations of the Navier-Stokes
equations in order to confront these predictions with the actual behaviour of
the bands. Computation of the autocorrelation function of the modulation of
turbulence shows quantitative agreement with the model: in the established band
regime, the amplitude of modulation follows an Ornstein-Uhlenbeck process. In
order to test the S3T predictions, we perform quench experiments, sudden
decreases of the Reynolds number from uniform turbulence, in which modulation
appears. We compute the average evolution of the amplitude of modulation and
the fluctuations around it. We find good agreement between numerics and
modeling. The average trajectory grows exponentially, at a rate clearly smaller
than that of the formation of laminar holes. The actual time evolution remains
in a flaring envelope, centred on the average, and expanding at the same rate.
These results provide further validation of the stochastic modeling for the
time evolution of the bands for further studies. They stress on the difference
between the oblique band formation and the formation of laminar holes.
