Eigenfunctions of the Multidimensional Linear Noise Fokker-Planck Operator via Ladder Operators

TL;DR

This paper develops ladder operators to explicitly derive eigenfunctions and eigenvalues of the multidimensional linear noise Fokker-Planck operator, extending known one-dimensional results to higher dimensions.

Contribution

It introduces raising and lowering operators for the multidimensional Fokker-Planck operator, providing explicit eigenfunctions and eigenvalues, and demonstrating their bi-orthogonality and relation to Hermite functions.

Findings

Eigenfunctions form a bi-orthogonal set.

Eigenfunctions reduce to sums of Hermite functions.

Explicit expressions for eigenvalues and eigenfunctions are derived.

Abstract

The eigenfunctions and eigenvalues of the Fokker-Planck operator with linear drift and constant diffusion are required for expanding time-dependent solutions and for evaluating our recent perturbation expansion for probability densities governed by a nonlinear master equation. Although well-known in one dimension, for multiple dimensions the eigenfunctions are not explicitly given in the literature. We develop raising and lowering operators for the Fokker-Planck (FP) operator and its adjoint, and use them to obtain expressions for the corresponding eigenvalues and eigenfunctions. We show that the eigenfunctions for the forward and adjoint FP operators form a bi-orthogonal set, and that the eigenfunctions reduce to sums of products of Hermite functions in a particular coordinate system.