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

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.