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On uniqueness of invariant measures for random walks on
Ergodic Theory and Dynamical Systems ( IF 0.9 ) Pub Date : 2021-04-29 , DOI: 10.1017/etds.2021.31
SARA BROFFERIO , DARIUSZ BURACZEWSKI , TOMASZ SZAREK

We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$ . In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was studied by Choquet and Deny [Sur l’équation de convolution $\mu = \mu * \sigma $ . C. R. Acad. Sci. Paris250 (1960), 799–801] in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on a broader class of systems satisfying the conditions of recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${\mathbb R}$ . Our work can be viewed as following on from Babillot et al [The random difference equation $X_n=A_n X_{n-1}+B_n$ in the critical case. Ann. Probab.25(1) (1997), 478–493] and Deroin et al [Symmetric random walk on $\mathrm {HOMEO}^{+}(\mathbb {R})$ . Ann. Probab.41(3B) (2013), 2066–2089].



中文翻译:

关于随机游走的不变测度的唯一性

我们考虑在实线${\mathbb R}$ 的方向保持同胚群上的随机游走 。特别是,提出了生成过程的不变度量的唯一性这一基本问题。Choquet 和 Deny [Sur l'équation de convolution $\mu = \mu * \sigma $ 研究了这个问题。CR学院。科学。巴黎250(1960), 799–801] 在由直线平移产生的随机游走的背景下。如今,在强收缩系统的一般设置中,答案已得到很好的理解。在这里,我们专注于满足循环、收缩和无限动作条件的更广泛的系统类别。 我们证明了在这些条件下,随机过程在${\mathbb R}$ 上具有唯一不变的 Radon 测度 。我们的工作可以看作是 Babillot等人的 [The random Difference equation $X_n=A_n X_{n-1}+B_n$ in the critical case。安。概率。25 (1) (1997), 478–493] 和 Deroin等人[ $\mathrm {HOMEO}^{+}(\mathbb {R})$ 上的对称随机游走 。安。概率。41 (3B) (2013), 2066–2089]。

更新日期:2021-04-29
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