Some remarks on degenerate hypoelliptic Ornstein-Uhlenbeck operators

TL;DR

This paper applies recent quadratic operator analysis to degenerate hypoelliptic Ornstein-Uhlenbeck operators, revealing spectral properties, resolvent behavior, and exponential convergence to equilibrium.

Contribution

It demonstrates how general quadratic operator results can be used to analyze degenerate hypoelliptic Ornstein-Uhlenbeck operators, providing new resolvent estimates and stability insights.

Findings

Spectrum can be highly unstable under perturbations

Resolvent norms can blow up away from the spectrum

Exponential return to equilibrium is established

Abstract

We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in $L^2$ 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.