Ergodic aspects of some Ornstein-Uhlenbeck type processes related to Levy processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Bertoin, Jean¹ 

[1] Univ Zurich, Inst Math, Winterthurerstr 190, CH-8057 Zurich, Switzerland

关键词: Ornstein-Uhlenbeck type process;    Stationarity;    Self-similar Markov process;    Levy process;    Exponential functional;    Darling-Kac theorem;   

DOI  :  10.1016/j.spa.2018.05.007

来源: Elsevier

【 摘 要 】

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 reserved.

【 授权许可】

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