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

Valentina Casarino,Paolo Ciatti,Peter Sjögren

doi:10.1007/s00013-021-01637-6

Abstract

We study the orthogonality of the generalized eigenspaces of an Ornstein–Uhlenbeck operator {mathscr {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 {mathscr {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 {mathscr {L}} may or may not be orthogonal.

Highlights

We discuss the orthogonality of the generalized eigenspaces associated to a general Ornstein–Uhlenbeck operator L in RN
The Ornstein–Uhlenbeck semigroup Ht t>0 generated by L is not assumed to be self-adjoint in L2(γ∞); here γ∞ denotes the unique invariant probability measure under the action of the semigroup, and will be specified later
The Ornstein–Uhlenbeck operator L admits a complete system of generalized eigenfunctions; see [8]

Summary

Introduction

We discuss the orthogonality of the generalized eigenspaces associated to a general Ornstein–Uhlenbeck operator L in RN .Recently, the authors started studying some harmonic analysis issues in a nonsymmetric Gaussian context [1–3]. Ornstein–Uhlenbeck operator, Generalized eigenspaces, Orthogonality, Gaussian measure. The Hermite decomposition implies, in particular, that each generalized eigenfunction of L with a nonzero eigenvalue is orthogonal to the space of constant functions, that is, to the kernel of L .

Results

Conclusion