Spread Out Random Walks on Homogeneous Spaces
Abstract
A measure on a locally compact group is called spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the -step distributions: They equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.
Cite
@article{arxiv.1910.00467,
title = {Spread Out Random Walks on Homogeneous Spaces},
author = {Roland Prohaska},
journal= {arXiv preprint arXiv:1910.00467},
year = {2023}
}
Comments
30 pages, 1 figure; conceptual improvements in Section 3.1, Example 3.7 corrected, other minor changes and corrections. This article has been published in a revised form in Ergodic Theory and Dynamical Systems. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. (C) The Author, 2020