The rate of escape of the most visited site of Brownian motion

TL;DR

This paper proves that the most visited site of a one-dimensional Brownian motion escapes to infinity faster than a certain rate involving ^{1/2} and a logarithmic factor, confirming a conjecture by Lifshits and Shi.

Contribution

It establishes the asymptotic escape rate of the most visited site of Brownian motion, confirming a conjecture and extending results to simple random walk.

Findings

The most visited site diverges faster than ^{1/2} divided by a logarithmic factor for gamma>1.

The result holds almost surely for Brownian motion and simple random walk.

The paper confirms a conjecture of Lifshits and Shi regarding the escape rate.

Abstract

Let $\{L^z_t\}$ be the jointly continuous local times of a one-dimensional Brownian motion and let $L^*_t=\sup_{z\in \mathbb R} L^z_t$. Let $V_t$ be any point $z$ such that $L^z_t=L^*_t$, a most visited site of Brownian motion. We prove that if $\gamma>1$, then\[\liminf_{t\to \infty} \frac{|V_t|}{\sqrt t/(\log t)^\gamma}=\infty, \qquad \mbox{a.s.}, \] with an analogous result for simple random walk. This proves a conjecture of Lifshits and Shi.