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)) t0 which drifts to ∞, namely U (t) := e −t X(e t − 1). We point out that U is always a (topo-logically) recurrent Markov process and identify its invariant measure in terms of the law of the exponential functionaî I := ∞ 0 exp(ˆ ξs)ds, wherê ξ is the dual of the real-valued Lévy process ξ related to X by the Lamperti transformation. This invariant measure is infinite (i.e. U is null-recurrent) if and only if ξ1 ∈ L 1 (P). In that case, we determine the family of Lévy processes ξ 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 ξ, namely V (t) := exp(ξt) t 0 exp(−ξs)ds + V (0) , and properties of time-substitutions based on additive functionals.
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