On the Jordan decomposition for a class of non-symmetric Ornstein-Uhlenbeck operators

TL;DR

This paper derives the Jordan canonical form for a specific class of non-symmetric Ornstein-Uhlenbeck operators with Jordan block drift matrices, providing explicit eigenfunctions and multiplicities for low dimensions.

Contribution

It explicitly computes the Jordan decomposition for these operators, including eigenfunctions and geometric multiplicities, which was previously not well-understood.

Findings

Explicit Jordan form for 2D case

Method for 3D case involving Hermite polynomials

Determination of geometric multiplicity of eigenvalues

Abstract

In this paper, we calculate the Jordan decomposition (or say, the Jordan canonical form) for a class of non-symmetric Ornstein-Uhlenbeck operators with the drift coefficient matrix being a Jordan block and the diffusion coefficient matrix being identity multiplying a constant. For the 2-dimensional case, we present all the general eigenfunctions by the induction. For the 3-dimensional case, we divide the calculating of the Jordan decomposition into several steps (the key step is to do the canonical projection onto the homogeneous Hermite polynomials, next we use the theory of systems of linear equations). As a by-pass product, we get the geometric multiplicity of the eigenvalue of the Ornstein-Uhlenbeck operator.