On the Jordan decomposition for a class of nonsymmetric OrnsteinUhlenbeck operators
Abstract
In this paper, we calculate the Jordan decomposition (or say, the Jordan canonical form) for a class of nonsymmetric OrnsteinUhlenbeck operators with the drift coefficient matrix being a Jordan block and the diffusion coefficient matrix being identity multiplying a constant. For the 2dimensional case, we present all the general eigenfunctions by the induction. For the 3dimensional 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 bypass product, we get the geometric multiplicity of the eigenvalue of the OrnsteinUhlenbeck operator.
 Publication:

arXiv eprints
 Pub Date:
 December 2012
 arXiv:
 arXiv:1212.1852
 Bibcode:
 2012arXiv1212.1852C
 Keywords:

 Mathematics  Probability;
 Mathematics  Functional Analysis;
 47A75;
 60H99;
 47F05;
 33C45