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

Abstract

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

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