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

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$$\underset{t\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\mathrm{\mid }{V}_t\mathrm{\mid }}{\sqrt{t}\mathrm{/}(\log t{)}^{\gamma }}=\mathrm{\infty },a\mathrm{.}s\mathrm{.},$$ with an analogous result for simple random walk. This proves a conjecture of Lifshits and Shi.