Stochastic Differential Equations with Applications to Random Harmonic Oscillators and Wave Propagation in Random Media

The two-time method is used to obtain an expansion, valid for $\varepsilon $ small and t large, of the vector solution $u( {t,\varepsilon } )$ of an abstract ordinary differential equation involving $\varepsilon $. The same method is used to get expansions of functions of u. The results are shown to apply to the solutions of stochastic equations. They are used to find the first two moments and the transition probability of the displacement of a harmonic oscillator with spring constant a random function of t. The result contains the condition for mean square stability due to Stratonovich. The results are also applied to one-dimensional wave propagation through a layer with refractive index a random function of position. They are used to find the mean square amplitude reflection and transmission coefficients, which are just the mean power reflection and transmission coefficients. A graph of the mean square transmission coefficient as a function of layer thickness is presented. The results are also compared with those obtainable by other methods.