Moments and Correlation Functions of Solutions of a Stochastic Differential Equation

Abstract

This paper shows how to obtain exact, closed-form expressions for various moments and correlation functions of the solutions of the stochastic, ordinary differential equation d2udz2+β02[1+ηT(z)]u=0,where T(z) is the so-called ``random telegraph'' wave and β02 and η are positive real constants. These moments and correlation functions are calculated by two different methods, one a phase space method and the other a matrix method familiar from optics. It is found that the moments are sums of exponentials. The first-order moments decay exponentially but the second-order moments grow exponentially. The correlation functions are also sums of exponentials and show that the solutions do not form a stationary process. An important application of these results is obtained in the problem of a plane electromagnetic wave normally incident on a randomly stratified dielectric plate. It is shown that, if 𝔖 is the amplitude transmission coefficient of the plate, then 〈1/SS*〉 can be expressed in terms of the second-order moments of the solutions and derivatives of solutions of the stochastic differential equation.