Open Access
2023 The rate of escape of the most visited site of Brownian motion
Richard F. Bass
Author Affiliations +
Electron. J. Probab. 28: 1-12 (2023). DOI: 10.1214/23-EJP916


Let {Ltz} be the jointly continuous local times of a one-dimensional Brownian motion and let Lt=supzRLtz. Let Vt be any point z such that Ltz=Lt, a most visited site of Brownian motion. We prove that if γ>1, then

lim inft|Vt|t(logt)γ=,a.s.,

with an analogous result for simple random walk. This proves a conjecture of Lifshits and Shi.


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Richard F. Bass. "The rate of escape of the most visited site of Brownian motion." Electron. J. Probab. 28 1 - 12, 2023.


Received: 19 November 2021; Accepted: 3 February 2023; Published: 2023
First available in Project Euclid: 8 February 2023

MathSciNet: MR4546025
zbMATH: 1517.60098
MathSciNet: MR4529085
Digital Object Identifier: 10.1214/23-EJP916

Primary: 60J55

Keywords: Brownian motion , favorite point , most visited site , Rate of escape

Vol.28 • 2023
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