On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator

TL;DR

This paper investigates the orthogonality properties of generalized eigenspaces of the Ornstein--Uhlenbeck operator in multi-dimensional space, revealing conditions under which these eigenspaces are orthogonal with respect to the invariant Gaussian measure.

Contribution

It establishes orthogonality of eigenspaces when the drift matrix has a single eigenvalue and provides examples showing non-orthogonality when multiple eigenvalues are present.

Findings

Eigenspaces are orthogonal if the drift matrix has only one eigenvalue.

Orthogonality may fail when the drift matrix has multiple eigenvalues.

Examples demonstrate both orthogonal and non-orthogonal cases.

Abstract

We study the orthogonality of the generalized eigenspaces of an Ornstein--Uhlenbeck operator $\mathcal L$ in $\mathbb{R}^N$, with drift given by a real matrix $B$ whose eigenvalues have negative real parts. If $B$ has only one eigenvalue, we prove that any two distinct generalized eigenspaces of $\mathcal L$ are orthogonal with respect to the invariant Gaussian measure. Then we show by means of two examples that if $B$ admits distinct eigenvalues, the generalized eigenspaces of $\mathcal L$ may or may not be orthogonal.