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 $(X(t){)}_{t\ge 0}$ which drifts to $\infty$, namely $U(t):={\mathrm{e}}^{-t}X({\mathrm{e}}^t-1)$. We point out that $U$ is always a (topologically) recurrent Markov process and identify its invariant measure in terms of the law of the exponential functional $\hat{I}:={\int }_0^{\infty }\exp ({\hat{\xi }}_s)\mathrm{d}s$, where $\hat{\xi }$ is the dual of the real-valued Lévy process $\xi$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if ${\xi }_1\notin L^1(\mathbb{P})$. In that case, we determine the family of Lévy processes $\xi$ for which $U$ fulfills the conclusions of the Darling-Kac theorem. Our approach relies crucially on another generalized Ornstein-Uhlenbeck process that can be associated to the Lévy process $\xi$, namely $V(t):=\exp ({\xi }_t)\left({\int }_0^t\exp (-{\xi }_s)\mathrm{d}s+V(0)\right)$, and properties of time-substitutions based on additive functionals.

This new version gives credits to earlier works in the literature that I first missed