Ergodic aspects of some Ornstein-Uhlenbeck type processes related to Lévy processes
Résumé
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 $U(t)\coloneqq \e^{-t}X(\e^{t}-1)$.
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 exponential functional $\hat I\coloneqq \int_0^{\infty}\exp(\hat\xi_s)\dd s$, where $\hat\xi$ is the dual of the real-valued L\'evy process $\xi$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if $\xi_1\not \in L^1(\pr)$. In that case, we determine the family of L\'evy processes $\xi$ for which $U$ fulfills the conclusions of the Darling-Kac theorem. Our approach relies crucially on a remarkable connection due to Patie \cite{Patie} with
another generalized Ornstein-Uhlenbeck process that can be associated to the L\'evy process $\xi$, and properties of time-substitutions based on additive functionals.
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