Computing eigenfunctions of the multidimensional Ornstein-Uhlenbeck operator

TL;DR

This paper explores methods for computing eigenfunctions of the multidimensional Ornstein-Uhlenbeck operator, addressing practical challenges in higher dimensions and proposing a new computational approach.

Contribution

It introduces a tractable method for computing eigenfunctions in general cases, extending beyond special cases with known solutions.

Findings

Reviewed special cases with explicit eigenfunctions

Proposed a new computational approach for general eigenfunctions

Discussed the dimension dependence of the method

Abstract

We discuss approaches to computing eigenfunctions of the Ornstein--Uhlenbeck (OU) operator in more than two dimensions. While the spectrum of the OU operator and theoretical properties of its eigenfunctions have been well characterized in previous research, the practical computation of general eigenfunctions has not been resolved. We review special cases for which the eigenfunctions can be expressed exactly in terms of commonly used orthogonal polynomials. Then we present a tractable approach for computing the eigenfunctions in general cases and comment on its dimension dependence.