Green's function for Markovian systems

Abstract

Many perturbation series solutions to stochastic differential equations suffer from secularity; that is, individual terms in the series behave as some power of an independent variable, and thus diverge as that variable goes to infinity. We derive a method of renormalizing the ``free'' or ``unperturbed'' Green's function by summing a certain class of terms and including them in the Green's function. The terms included in this class correspond to Markovian interaction with the stochastic field. The remainder of the perturbation series corresponds to non-Markovian corrections. We give a diagrammatic interpretation of the individual terms in the perturbation series. We may solve, under certain assumptions, the explicit form of the Markovian Green's function. We apply this method to plasma turbulence, and show that it produces Dupree's theory of turbulence. We derive the condition for validity of the perturbation expansion, which is that particles are not ``trapped'' by the turbulent waves.