We study the orthogonality of the generalized eigenspaces of an Ornstein– Uhlenbeck operator L in R^N , with drift given by a real matrix B whose eigenvalues have negative real parts. If B has only one eigenvalue, we prove that any two distinct generalized eigenspaces of L are orthogonal with respect to the invariant Gaussian measure. Then we show by means of two examples that if B admits distinct eigenvalues, the generalized eigenspaces of L may or may not be orthogonal.
On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator
VALENTINA CASARINO
;PAOLO CIATTI;
2021
Abstract
We study the orthogonality of the generalized eigenspaces of an Ornstein– Uhlenbeck operator L in R^N , with drift given by a real matrix B whose eigenvalues have negative real parts. If B has only one eigenvalue, we prove that any two distinct generalized eigenspaces of L are orthogonal with respect to the invariant Gaussian measure. Then we show by means of two examples that if B admits distinct eigenvalues, the generalized eigenspaces of L may or may not be orthogonal.
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