Generalized Fokker-Planck equation: Derivation and exact solutions

TL;DR

This paper derives a generalized Fokker-Planck equation for overdamped particles under multiplicative noise with arbitrary distributions, providing exact solutions for specific noise types and potentials.

Contribution

It introduces a unified derivation of the generalized Fokker-Planck equation for arbitrary noise distributions and solves it exactly for key potentials.

Findings

Reproduces known Fokker-Planck equations for Poisson and Lévy noises.

Provides explicit analytical solutions for linear, quadratic, and tailored potentials.

Analyzes time-dependent and stationary behaviors of the solutions.

Abstract

We derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise generating process. We explicitly consider this equation for various specific types of noises, including Poisson white noise and L\'{e}vy stable noise, and show that it reproduces all Fokker-Planck equations that are known for these noises. Exact analytical, time-dependent and stationary solutions of the generalized Fokker-Planck equation are derived and analyzed in detail for the cases of a linear, a quadratic, and a tailored potential.