Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Ergodic aspects of some Ornstein-Uhlenbeck type processes related to Lévy processes (1706.08421v2)

Published 26 Jun 2017 in math.PR

Abstract: This work concerns the Ornstein-Uhlenbeck type process associated to a positive self-similar Markov process $(X(t))_{t\geq 0}$ 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 $\hat I := \int_0\infty \exp(\hat\xi_s) {\rm d}s$, where $\hat\xi$ 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 $\xi_1\not \in L1(\mathbb{P})$. 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 $V(t) := \exp(\xi_t)\left(\int_0t \exp(-\xi_s){\rm d}s +V(0)\right)$, and properties of time-substitutions based on additive functionals.

Summary

We haven't generated a summary for this paper yet.