Analytical approximation to the multidimensional Fokker--Planck equation with steady state

TL;DR

This paper develops an asymptotic approximation method for solving the multidimensional Fokker--Planck equation, providing explicit formulas that are computationally efficient and accurate across various complex systems.

Contribution

It introduces a novel asymptotic approach based on the Ornstein--Uhlenbeck case to approximate solutions of the multidimensional Fokker--Planck equation, applicable to complex potentials.

Findings

Explicit formulas are fast to evaluate.

Approximation performs well even for non-OU systems.

Works effectively in both 1D and 2D cases.

Abstract

The Fokker--Planck equation is a key ingredient of many models in physics, and related subjects, and arises in a diverse array of settings. Analytical solutions are limited to special cases, and resorting to numerical simulation is often the only route available; in high dimensions, or for parametric studies, this can become unwieldy. Using asymptotic techniques, that draw upon the known Ornstein--Uhlenbeck (OU) case, we consider a mean-reverting system and obtain its representation as a product of terms, representing short-term, long-term, and medium-term behaviour. A further reduction yields a simple explicit formula, both intuitive in terms of its physical origin and fast to evaluate. We illustrate a breadth of cases, some of which are `far' from the OU model, such as double-well potentials, and even then, perhaps surprisingly, the approximation still gives very good results when compared with numerical simulations. Both one- and two-dimensional examples are considered.