Conditioned random walks on linear groups I: construction of the target harmonic measure

• 成果类型: Article; Early Access
• 署名作者: Grama, Ion; Quint, Jean-Francois; Xiao, Hui
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Montpellier; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01508-7
• 发表日期: 2026-06-09
• 关键词: Random walks on groups; exit time; Random walks conditioned to stay positive; Local limit theorem; Target harmonic measure; local limit-theorem; probabilities

摘要

Our objective is to explore random walks on the general linear group, constrained to a specific domain, with a primary focus on establishing the conditioned local limit theorem. This paper represents the first step toward achieving this goal, specifically entailing the construction of a novel entity - the target harmonic measure. This measure, together with the harmonic function, serves as a pivotal component in establishing the conditioned local limit theorem. Using a reversal identity, we introduce a reversed sequence characterized as a dual random walk with a perturbation depending on future observations. The investigation of such walks, which rely on future information, lies at the heart of this paper. To carry out this study, we develop an approach grounded in the finite-size approximation of perturbations, enabling us to simplify the investigation to an array of Markov chains with increasing dimensions.