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 $\mathit{\gamma }>1$, then

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{|V_t|}{\sqrt{t}∕{(logt)}^{\mathit{\gamma }}}=\mathrm{\infty },\text{a.s.},$$

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