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arXiv · 2103.09698

On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator

Valentina Casarino·Paolo Ciatti·Peter Sjögren
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Abstract

We study the orthogonality of the generalized eigenspaces of an Ornstein--Uhlenbeck operator $\mathcal L$ in $\mathbb{R}^N$, 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 $\mathcal L$ 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 $\mathcal L$ may or may not be orthogonal.

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Valentina Casarino, Paolo Ciatti, Peter Sjögren. 2021-03-17. On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator. https://doi.org/10.1007/s00013-021-01637-6

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