Jordan decomposition and geometric multiplicity for a class of non-symmetric Ornstein-Uhlenbeck operators
© Rao et al.; licensee Springer. 2014
Received: 8 November 2013
Accepted: 6 January 2014
Published: 27 January 2014
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.
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 so-called complex Hermite polynomials  (or, say, the Hermite-Laguerre-Itô polynomials) where the idea is to proceed by means of a decomposition to the summation of series of up to a 2-dimensional normal Ornstein-Uhlenbeck operator . But if the Ornstein-Uhlenbeck operator is not normal, the general eigenfunctions are still unknown up to now.
where is the diffusion coefficient matrix and is the transpose matrix of B.
In the present paper, we present an approach to calculate the Jordan decomposition and the generalized eigenfunctions (see Theorems 2.1, 3.1) for .b The proof of Theorem 2.1 is by direct calculation. The main techniques of the proof of Theorem 3.1 are canonical projection and the theory of systems of linear equations. This approach is novel to the Jordan decomposition of differential operators as far as we know.
It is a difficult problem to get the geometric multiplicity of the eigenvalue of a differential operator from the perspective of functional analysis. It is well known that the spectrum of the symmetric Ornstein-Uhlenbeck operator is the starting point of stochastic analysis (more precisely: Malliavin calculus), thus the analogous results of the non-symmetric Ornstein-Uhlenbeck operator are interesting. We will treat the more general non-symmetric Ornstein-Uhlenbeck operator in the future.
2 In case of dimension 2
where and when . In particular, is the eigenfunction associated to γ.
3 In case of dimension 3
Notation 1 Set , and .
the Hermite polynomials are another basis of .
We denote by the canonical projection  of onto .
The proof of Theorem 3.1 is presented in Section 3.1. The following is a by-product.
Corollary 3.2 The geometric multiplicity of the eigenvalue γ of the Ornstein-Uhlenbeck operator is .
3.1 Proof of Theorem 3.1
For convenience, Eq. (3.5) can be rewritten in the following way.
where we denote by the triple integers the Hermite polynomial . It follows from Eq. (3.9) that the weights of the arrows between and , are −j, −i, respectively.
One can find many properties from the directed acyclic graph. For example, for the vertex with , the height (which is defined by the distance between vertices and and thus is between ) is . For simplicity, in each height of the graph, we list the vertices decreasingly by lexicographic order. Then the vertices and are symmetric about the n th height of the graph.
It follows from Eq. (1.4), Corollary 3.2, and Theorem 3.1 that the order of the graph (the numbers of vertices in the graph) is the algebraic multiplicity of γ, are 1 plus the distance between vertices and , and the numbers of vertices in the n th height of the graph is the geometric multiplicity of γ.
Proposition 3.4 The spectral subspace associated to belongs to .
Proof Suppose that f is a generalized eigenfunction, i.e., there exists an integer such that . It follows from Eq. (3.7) that if the degree of f is , then . This is a contradiction; then and the degree is exactly n.d If there is an such that , then by Eq. (3.6), . This is a contradiction; then . □
Lemma 3.5 For any polynomial with , there exists a unique solution to the equation .
By the linear independence of , we have a system of linear equations in unknowns.
We sort , which appear in the directed acyclic graph in Remark 1. The coefficient matrix of the linear equations is a lower triangle matrix with nonzero diagonal entry . Thus the linear equations have a unique solution. □
Clearly, if f satisfies , so does where h is any eigenfunction of associated to γ. Thus we have the following proposition.
Proposition 3.6 For any with , , there exist solutions to the equation . In addition, if f is the same degree polynomial to g then there exists one and only one solution.
Proposition 3.7 Set , . Then it satisfies the equation .
Clearly, the solution is 1-dimensional and , , is a solution. □
Corollary 3.8 The geometric multiplicity of the eigenvalue is greater than or equal to (i.e., there are at least independent solutions to the equation ).
Denote by , the solutions to the equation given by Corollary 3.8. Clearly, , where is the same as in Proposition 3.7.
Note that . Set , then the l.h.s. of Eq. (3.14) is a linear mapping from the to the . The linear mapping is the evolution from the 2k-height to the -height of the directed acyclic graph in Remark 1, which is represented in the natural basis by a -square matrix (it is the multiplication of some matrices; for details, please refer to Section 3.2). By Proposition 3.10, the matrix is nonsingular, which implies that Eq. (3.14) has a solution. Since , it follows from Proposition 3.6 that Eq. (3.15) has a solution. Then we have the following proposition.
Proposition 3.9 There exists an such that .
Proof of Theorem 3.1 Note that in Proposition 3.7 are linear independent, so are the eigenfunctions . Let be as in Proposition 3.9. Then the generalized eigenfunctions are linear independent (please refer to the proof of [, p.264, Theorem 6.2.]).
Thus forms a basis of the spectral subspace associated to γ. Together with Proposition 3.8, we see that the geometric multiplicity of the eigenvalue γ is equal to (otherwise, the algebraic multiplicity should be greater than ). Then forms the basis of the eigenspace of γ. Equation (3.4) is exactly the conclusion of Proposition 3.7. □
3.2 Linear mapping represented by the multiplication of some matrices
Then the matrices associated to the linear mappings are , (see below).
The others are similar. In general, the matrix (see Proposition 3.9) associated to the linear mapping of the evolution from the -height to the -height of the directed acyclic graph is as follows.
Then the matrices are nonsingular.
Apply the notation in [10, 13]. Let A be an matrix, , and , , . Denote by , or simply , the submatrix of A consisting of the entries in rows and columns . Let be the submatrix of A obtained by deleting rows and columns . For convenience, () means deleting rows (columns) only.
Proof We divide the proof into three steps.
Claim 1: All the minors (i.e., the determinant of the square submatrix) of the matrices Ak, , , , , are nonnegative. Since () and () have the same types, we only need to prove the cases of , . The square submatrix of is the same as that of a certain , , which is a direct sum of two matrices  with the same type of or . Thus we only need to prove the case of . In fact, is a direct sum  of several nonnegative triangular matrices, which implies that has a nonnegative determinant.
Together with Claim 1 and by mathematical induction, we have .
Clearly, . By the induction assumption , Claim 2 implies that there exists at least one . This ends the proof. □
a It is well known that the reversibility is equivalent to the symmetric property of the generator for a Markov process with invariant distribution.
b We think that the approach is also able to solve the same problems for .
c The directed acyclic graph can easily be given for any integer n, and we ignore it for convenience.
d The reader can also refer to [, Proposition 3.1].
This work was supported by NSFC (No. 11101137).
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