Construction of an exact solution of the Dyson equation for the mean value of the Green's function

A solution is found for the mean value of the Green's function of a stochastic linear system of general form with Gaussian fluctuating parameters. The method is based on constructing higher approximations of the Dyson equation by closing the chains of equations for the mean values of the variational derivatives of the solution at a certain step. It is shown that for the case of exponentially correlated fluctuations of the parameters of the system, the exact solution of the Dyson equation can be represented as an infinite continued fraction. The results are illustrated by the finding of the dynamical characteristics of an harmonic oscillator which has fluctuations of the eigenfrequency and the losses.