Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes

Abstract

This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process $( X ( t ) )_{t \geq 0}$ which drifts to $\infty$, namely $U ( t )$= $e^{-1} X (e^t - 1 )$ . We point out that $U$ is always a (topologically) recurrent ergodic Markov process. We identify its invariant measure in terms of the law of the exponentialfunctional $\hat{I} $ = $\int\limits_0^\infty exp ( \hat{\varepsilon}) ds $, where $\hat{\varepsilon}$ is the dual of the real-valued Lévy process $\varepsilon$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if $\varepsilon_1 \notin L^1 (\mathbb{P} )$ . In that case, we determine the family of Lévy processes $\varepsilon$ for $U$ which fulfills the conclusions of the Darling–Kac theorem. Our approach relies crucially on a remarkable connection due to Patie (Patie, 2008) with another generalized Ornstein–Uhlenbeck process that can be associated to the Lévy process $\varepsilon$, and properties of time-substitutions based on additive functionals.