  We analyse the asymptotic growth of the error for Hamiltonian flows due to
small random perturbations. We compare the forward error with the reversibility
error, showing their equivalence for linear flows on a compact phase space. The
forward error, given by the root mean square deviation $\sigma(t)$ of the noisy
flow, grows according to a power law if the system is integrable and according
to an exponential law if it is chaotic.
  The autocorrelation and the fidelity, defined as the correlation of the
perturbed flow with respect to the unperturbed one, exhibit an exponential
decay as $\exp\left(-\sigma^2(t)\right)$. Some numerical examples such as the
anharmonic oscillator and the H\'enon Heiles model confirm these results. We
finally consider the effect of the observational noise on an integrable system,
and show that the decay of correlations can only be observed after a sequence
of measurements and that the multiplicative noise is more effective if the
delay between two measurements is large.
