Persistence of Gaussian processes: non-summable correlations

• 成果类型: Article
• 署名作者: Dembo, Amir; Mukherjee, Sumit
• 署名单位: Stanford University; Columbia University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0746-9
• 发表日期: 2017
• 页码: 1007-1039
• 关键词: phi interface model; random polynomials; large deviations; fluctuating interfaces; stationary-processes; brownian-motion; probabilities; exponents; equation; maximum

摘要

Suppose the auto-correlations of real-valued, centered Gaussian process are non-negative and decay as for some regularly varying at infinity of order . With its primitive, we show that the persistence probabilities decay rate of is precisely of order , thereby closing the gap between the lower and upper bounds of Newell and Rosenblatt (Ann. Math. Stat. 33:1306-1313, 1962), which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of Sakagawa (Adv. Appl. Probab. 47:146-163, 2015) about the dependence on d of such persistence decay for the Langevin dynamics of certain -interface models on Z(d).