This work concerns the Ornstein-Uhlenbeck type process associated to a positive self-similar
Markov process (X(t))(t >= 0) which drifts to infinity, namely U(t) := 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
(I) over cap := integral(infinity)(0)exp((xi) over cap (s))ds, where (xi) over cap
is the dual of the real-valued Levy 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)
is not an element of L-1 (P). In that case, we determine the family of Levy processes
xi for which U 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 Levy process xi, and properties
of time-substitutions based on additive functionals. (C) 2018 Elsevier B.V. All rights