Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension

• 成果类型: Article
• 署名作者: Prigent, Martin; Roberts, Matthew I.
• 署名单位: University of Bath
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00978-7
• 发表日期: 2020
• 页码: 327-367
• 关键词: invariance

摘要

We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension 1/2 almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after n steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence epsilon(n) such that n epsilon(n) -> infinity. This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.