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On uniqueness of invariant measures for random walks on
${\textup {HOMEO}}^+(\mathbb R)$
Published online by Cambridge University Press: 29 April 2021
Abstract
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].
MSC classification
- Type
- Original Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
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