arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1706.08421math.PR
收藏

与Lévy过程相关的Ornstein-Uhlenbeck型过程的遍历性

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

Jean Bertoin

备注：
This new version gives credits to earlier works in the literature that I first missed
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文研究了正自相似马尔可夫过程$(X(t))_{t\geq 0}$漂移到$\infty$的Ornstein-Uhlenbeck型过程，即$U(t):= {\rm e}^{-t}X({\rm e}^t-1)$ .指出$U$始终是一个（拓扑）递归Markov过程，并根据指数泛函$\hat I := \int_0^\infty \exp(\hatξ_s) {\rm d}s$的规律确定了它的不变测度，其中$\hatξ$是实值Lévy过程$ξ$通过Lamperti变换与$X$相关的对偶.这个不变测度是无限的（即$U$是零递归的）当且仅当$ξ_1\not \in L^1(\mathbb{P})$。在这种情况下，我们确定了Lévy过程族$ξ$，其中$U$满足Darling-Kac定理的结论。我们的方法依赖于另一个广义的Ornstein-Uhlenbeck过程，可以关联到Lévy过程$ξ$，即$V(t) := \exp(ξ_t)\left(\int_0^t \exp(-ξ_s){\rm d}s +V(0)\right)$，和基于加性泛函的时间替换的性质。

英文摘要：

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 $\hat I := \int_0^\infty \exp(\hatξ_s) {\rm d}s$, where $\hatξ$ 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 $ξ_1\not \in L^1(\mathbb{P})$. 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 $V(t) := \exp(ξ_t)\left(\int_0^t \exp(-ξ_s){\rm d}s +V(0)\right)$, and properties of time-substitutions based on additive functionals.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑