Theory of nonlinear Langevin equation with quadratic noise

A generalized Langevin equation with a multiplicative linear and quadratic Ornstein–Uhlenbeck stochastic noise is investigated. An exact generalized Fokker–Planck equation for this Langevin equation is obtained. This generalized Fokker–Planck equation has the form of an integro–differential equation. An exact form of the integral kernel is given in terms of an operator-valued matrix-continued fraction. In the limit of small fluctuations and short coherence time of the noise a standard Fokker–Planck equation with drift and diffusion is derived in a systematic way.