Variational Schemes in the Fokker-Planck Equation

Abstract

We investigate variational methods for finding approximate solutions to the Fokker-Planck equation, especially in cases lacking detailed balance. These schemes fall into two classes: those in which a Hermitian operator is constructed from the (non-Hermitian) Fokker-Planck operator, and those which are based on soluble approximations to this operator. The various approaches are first tested on a simple quantum-mechanical problem and then applied to a toy Fokker-Planck equation. The problem of a particle moving in a potential and subject to external non-white noise is then investigated using the formalism developed earlier on in the paper.