Abstract
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.
Highlights
For the symmetric Ornstein-Uhlenbeck operator, the eigenfunctions are the well-known Hermite polynomials [ ]
The eigenfunctions of a type of finite-dimensional normal but non-symmetric Ornstein-Uhlenbeck operators have recently been found. They are the socalled complex Hermite polynomials [ ] where the idea is to proceed by means of a decomposition to the summation of series of up to a -dimensional normal Ornstein-Uhlenbeck operator [ ]
We present an approach to calculate the Jordan decomposition and the generalized eigenfunctions for d =, .b The proof of Theorem . is by direct calculation
Summary
1 Introduction For the symmetric Ornstein-Uhlenbeck operator, the eigenfunctions are the well-known Hermite polynomials [ ]. The geometric multiplicity of the eigenvalue γ of the Ornstein-Uhlenbeck operator A is r + . It follows from the property of the Hermite polynomials [ ] that (γ – A ) Hi(x)Hj(y)Hk(z) = (m – n)cHi(x)Hj(y)Hk(z) – iHi– (x) Hj+ (y) + jρHj– (y) Hk(z) – jHi(x)Hj– (y) Hk+ (z) + kρHk– (z) .
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