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

Bertoin, Jean (2019). Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes. Stochastic Processes and their Applications, 129(4):1443-1454.

## 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.

## 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.

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