Calculation of Correlation Functions of Solutions of a Stochastic Ordinary Differential Equation

In this paper we use the ``smoothing method'' to calculate the correlation functions of the solutions of the equation d2udz2+β02[1+ηN(z)]u=0,satisfying nonstochastic initial conditions, where N(z) is a real, wide-sense stationary stochastic process with zero mean and β0 and η ≪ 1 are positive constants. It is shown that an appropriate application of the smoothing method leads to the exact results in the case when N(z) is the random telegraph process. Moreover, under appropriate conditions on the general process N(z), approximate expressions are obtained for the correlation functions in terms of the first- and second-order moments of the solutions, and approximate expressions are given for these moments.