64 - Fokker—Planck Equation

Abstract

This chapter reviews the solution of Fokker–Planck equation applicable to linear ordinary differential equations with linearly appearing “white Gaussian noise” terms. If a differential equation contains random terms, then the solution to the differential equation can only be described statistically. The solution to the Fokker–Planck equation is the probability density of the solution to the original differential equation. The chapter presents the technique for constructing the Fokker–Planck equation for a linear system of ordinary differential equations depending on several white noise terms. The chapter highlights that with a “Fourier” transform, the method of characteristics can often solve a Fokker–Planck equation in one dimension. Because a Fokker–Planck equation and the equation for a Green's function both have delta function forcing terms, the solution techniques are similar. Not all noise terms are white Gaussian noise. When the coefficient of the noise term is small, then a singular perturbation problem generally results.