A perturbative approach to a class of Fokker-Planck equations

TL;DR

This paper introduces a perturbative method for solving specific Fokker-Planck equations with constant diffusion and small drift parameters, leveraging their connection to Schrödinger equations, demonstrated through two illustrative examples.

Contribution

The paper presents a novel perturbative approach linking Fokker-Planck and Schrödinger equations to approximate solutions for equations with small drift terms.

Findings

Effective for equations with time-dependent drift

Applicable to Uhlenbeck-Ornstein process with small drift

Provides approximate solutions using perturbation theory

Abstract

In this paper we present a direct perturbative method to solving certain Fokker-Planck equations, which have constant diffusion coefficients and some small parameters in the drift coefficients. The method makes use of the connection between the Fokker-Planck and Schr\"odinger equations. Two examples are used to illustrate the method. In the first example the drift coefficient depends only on time but not on space. In the second example we consider the Uhlenbeck-Ornstein process with a small drift coefficient. These examples show that the such perturbative approach can be a useful tool to obtain approximate solutions of Fokker-Planck equations with constant diffusion coefficients.