Langevin treatment of the Fokker–Planck process

Abstract

This chapter discusses the Langevin treatment of the Fokker–Planck process and diffusion. The form of Langevin equation used is different from the stochastic differential equation using Ito's calculus lemma. The transform of the Langevin equation obeys the ordinary calculus rule, hence can be easily performed and some misleadings can be avoided. The origin of the difference between this approach and that using Ito's lemma comes from the different definitions of the stochastic integral. This chapter also discusses drift velocity, an example with an exact solution, use of Langevin equation for a general random variable, extension of this equation to the multiple dimensional case, and means of products of random variables and noise source.