We analyze the statistical behavior of signals in nonlinear circuits with delayed
feedback in the presence of external Markovian noise. For the special class of
circuits with intense phase mixing we develop an approach for the computation
of the probability distributions and multitime correlation functions based on the
random phase approximation. Both Gaussian and Kubo-Andersen models of
external noise statistics are analyzed and the existence of the stationary
(asymptotic) random process in the long-time limit is shown. We demonstrate
that a nonlinear system with chaotic behavior becomes a noise amplifier with
specific statistical transformation properties.