The most visited sites of symmetric stable processes |
| |
Authors: | Richard F Bass Nathalie Eisenbaum Zhan Shi |
| |
Institution: | (1) Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009 USA. e-mail: bass@math.uconn.edu, US;(2) Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France. e-mail: nae@ccr.jussieu.fr; e-mail: zhan@proba.jussieu.fr, FR |
| |
Abstract: | Let X be a symmetric stable process of index α∈ (1,2] and let L
x
t
denote the local time at time t and position x. Let V(t) be such that L
t
V(t)
= sup
x∈
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
→∞ |V(t)| = ∞ 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.
Received: 14 October 1998 / Revised version: 8 June 1999 / Published online: 7 February 2000 |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|