Propagation of Disturbances in a Random Medium

In this paper we are concerned with the determination of thestochastic rays along which light disturbances propagate ina random inhomogeneous, time-dependent and isotropic medium.The analysis is formulated in the framework of stochastic optimalcontrol. We state an appropriate stochastic Fermat's Principlewhich upon invoking the principle of optimality in dynamic programmingleads to a parabolic functional differential equation for thetraversal time of a wavefront, the randomness entering as anadditive white-noise in the displacements of the field. Whenthe randomness constitutes small perturbations to the mean velocityfield, the problem becomes one of singular perturbation of theHamilton-Jacobi equation by a small second order term. Approximateexpressions are presented for the traversal time and mean stochasticpath for a time-independent mean-velocity of propagation. Asa specific example we consider these expressions for propagationsin a stratified medium.