Local limit theorems for random walks on nilpotent Lie groups

• 成果类型: Article
• 署名作者: Benard, Timothee; Breuillard, Emmanuel
• 署名单位: Centre National de la Recherche Scientifique (CNRS); University of Oxford
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01387-4
• 发表日期: 2025
• 页码: 875-947
• 关键词: ergodic averages; distributions; equidistribution; Orbits; GROWTH; number

摘要

We establish the (non-lattice) local limit theorem for products of i.i.d. random variables on an arbitrary simply connected nilpotent Lie group G, where the variables are allowed to be non-centered. Our result also improves on the known centered case by proving uniformity for two-sided moderate deviations and allowing measures with a moment of order 2(dimG)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2(\dim G)<^>2$$\end{document} without further regularity assumptions. As applications we establish a Ratner-type equidistribution theorem for unipotent walks on homogeneous spaces and obtain a new proof of the Choquet-Deny property in our setting.