  Interferometric gravitational wave detectors are devoted to pick up the
effect induced on masses by gravitational waves. The variations of the length
dividing two mirrors is measured through a laser interferometric technique. The
Brownian motion of the masses related to the interferometer room temperature is
a limit to the observation of astrophysical signals. It is referred to as
thermal noise and it affects the sensitivity of both the projected and the
future generation interferometers. In this paper we investigate the relevance
of small non-linear effects and point out their impact on the sensitivity curve
of interferometric gravitational wave detectors (e.g. VIRGO, LIGO, GEO, ...)
through perturbative methods and numerical simulations. We find that in the
first order approximation the constants characterizing the power spectrum
density (PSD) are renormalized but it retains its typical shape. This is due to
the fact that the involved Feynman diagrams are of tadpole type. Higher order
approximations are required to give rise to up-conversion effects. This result
is predicted by the perturbative approach and is in agreement with the
numerical results obtained by studying the system's non-linear response by
numerically simulating its dynamics.
