Solution of Fokker-Planck equation for a broad class of drift and diffusion coefficients

TL;DR

This paper derives exact solutions to the Fokker-Planck equation for a class of Langevin equations with separable, variable drift and diffusion coefficients, providing insights into systems with spatially dependent diffusion.

Contribution

It introduces a class of exact solutions for the Fokker-Planck equation with variable drift and diffusion coefficients in the Stratonovich framework.

Findings

Derived explicit solutions for specific drift and diffusion functions.

Analyzed the impact of spatially dependent diffusion coefficient D(x).

Provided mathematical insights into variable-coefficient stochastic systems.

Abstract

We consider a Langevin equation with variable drift and diffusion coefficients separable in time and space and its corresponding Fokker-Planck equation in the Stratonovich approach. From this Fokker-Planck equation we obtain a class of exact solutions with the same spatial drift and diffusion coefficients. Furthermore, we analyze some details of this system by using the spatial diffusion coefficient $D(x)=\sqrt{D}|x| ^{-% \frac{\theta}{2}}$.