Some remarks on degenerate hypoelliptic Ornstein–Uhlenbeck operators

Abstract

We study degenerate hypoelliptic Ornstein–Uhlenbeck operators in L2 spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein–Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein–Uhlenbeck operators may be directly recovered from the general analysis of quadratic operators with zero singular spaces. We also provide new resolvent estimates for hypoelliptic Ornstein–Uhlenbeck operators. We show in particular that the spectrum of these non-selfadjoint operators may be very unstable under small perturbations and that their resolvents can blow-up in norm far away from their spectra. Furthermore, we establish sharp resolvent estimates in specific regions of the resolvent set which enable us to prove exponential return to equilibrium.