Quasilinear approximations for the propagator of the Fokker-Planck equation

Abstract

Four novel classes of the approximate Fokker-Planck propagators are developed by means of a decoupling operator treatment. For a small time increment t these are correct at least through second order t. The propagators have an Ornstein-Uhlenbeck form, thus leading to novel discrete path integrals with a linear reference system. Their ambiguities are characterized by an arbitrary function of the time increment, and by a matrix and a vector field which may be both time- and state-dependent. The efficiency of the treatment used is very sensitive to the choice of these functions, several forms of which are proposed. The power of the treatment is illustrated by a physically meaningful Fokker-Planck equation relevant to laser physics. Being used as the global approximations, the approximate propagators obtained are shown to provide a correct description of a bistable system in the entire time domain. A scaling theory of Suzuki, including its unified version, emerges from them in a very natural way. Another attractive feature of our general treatment when compared to the others known in the literature, is that it allows for equations with singular diffusion matrices, two of which, a Kramers equation and a colored-noise problem, are considered.