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

Errors, Correlations and Fidelity for noisy Hamilton flows. Theory and numerical examples