The most visited sites of symmetric stable processes

• 成果类型: Article
• 署名作者: Bass, RF; Eisenbaum, N; Shi, Z
• 署名单位: University of Connecticut; Sorbonne Universite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050255
• 发表日期: 2000
• 页码: 391-404
• 关键词: sample path properties; MARKOV-PROCESSES; brownian-motion

摘要

Let X be a symmetric stable process of index alpha is an element of (1, 2] and let L-t(x) denote the local time at time t and position x. Let V(t) be such that L-t(V(t)) = sup(x is an element of R) L-t(x). We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim(t-->infinity) \V(t)\ = infinity almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension of the Ray-Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and further to the winding problem for planar Brownian motion.