On separable Fokker-Planck equations with a constant diagonal diffusion matrix

TL;DR

This paper classifies and solves certain (1+3)-dimensional Fokker-Planck equations with constant diagonal diffusion matrices that are separable, identifying conditions on drift coefficients and constructing explicit solutions.

Contribution

It provides a complete classification of separable Fokker-Planck equations with constant diagonal diffusion and derives conditions on drift coefficients for separability.

Findings

Drift coefficients must be linear for separability.

Necessary conditions for R-separability are established.

Explicit solutions with separated variables are constructed.

Abstract

We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients $B_1(\vec x),B_2(\vec x),B_3(\vec x)$ providing separability of the corresponding Fokker-Planck equations and carry out variable separation in the latter. It is established, in particular, that the necessary condition for the Fokker-Planck equation to be separable is that the drift coefficients $\vec B(\vec x)$ must be linear. We also find the necessary condition for R-separability of the Fokker-Planck equation. Furthermore, exact solutions of the Fokker-Planck equation with separated variables are constructed