Higher-Order Moments of the Inverse of a Linear Stochastic Operator

Abstract

A procedure is presented that allows the derivation of a single deterministic equation on each of the moments of the inverse of a linear stochastic operator. Applied to the two lowest moments, the procedure results in the Dyson equation and the Bethe–Salpeter equation, respectively. In terms of an operator notation, specific forms are given for equations on the lowest four moments. These forms can be easily generalized to obtain the equation governing any of the higher-order moments. In the case of gaussian statistics, the equations to be satisfied by the third- and fourth-order moments are extensions of equations presented by Molyneux. Molyneux used a modified diagrammatic technique and restricted his attention to small-angle scattering through a gaussian medium. Finally, the formulism is applied to forward-scattering propagation through a weakly inhomogeneous medium.