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- W4300463395 abstract "This work concerns the Ornstein-Uhlenbeck type process associated to a positive self-similar Markov process $(X(t))_{tgeq 0}$ which drifts to $infty$, namely $U(t):= {rm e}^{-t}X({rm e}^t-1)$. 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 $hat I := int_0^infty exp(hatxi_s) {rm d}s$, where $hatxi$ 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 $xi_1not in L^1(mathbb{P})$. 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 another generalized Ornstein-Uhlenbeck process that can be associated to the L'evy process $xi$, namely $V(t) := exp(xi_t)left(int_0^t exp(-xi_s){rm d}s +V(0)right)$, and properties of time-substitutions based on additive functionals." @default.
- W4300463395 created "2022-10-03" @default.
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- W4300463395 date "2017-09-20" @default.
- W4300463395 modified "2023-09-27" @default.
- W4300463395 title "Ergodic aspects of some Ornstein-Uhlenbeck type processes related to Lévy processes" @default.
- W4300463395 hasPublicationYear "2017" @default.
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