# Jordan decomposition and geometric multiplicity for a class of non-symmetric Ornstein-Uhlenbeck operators

Research paper by Yulei Rao, Jiying Wang, Yong Chen

Indexed on: 27 Jan '14Published on: 27 Jan '14Published in: Advances in difference equations

#### Abstract

In this paper, we calculate the Jordan decomposition for a class of non-symmetric Ornstein-Uhlenbeck operators with the drift coefficient matrix, being a Jordan block, and the diffusion coefficient matrix, being the identity multiplying a constant. For the 2-dimensional case, we present all the general eigenfunctions by mathematical induction. For the 3-dimensional case, we divide the calculation of the Jordan decomposition into three steps. The key step is to do the canonical projection onto the homogeneous Hermite polynomials, and then use the theory of systems of linear equations. Finally, we get the geometric multiplicity of the eigenvalue of the Ornstein-Uhlenbeck operator.