How long does it take to see a flat Brownian path on the average? (Q1210124)

Consider a standard Brownian motion and define \(R(t,1)\) as its oscillation over the time interval \([t-1,1]\). Given \(\varepsilon>0\), let \(\tau(\varepsilon)\) be the smallest \(t\geq 1\) for which \(R(t,1)\leq\varepsilon\). The following results are proved: \[ \lim_{\varepsilon\to 0} \varepsilon^ 2 \log E(\tau(\varepsilon))=\pi^ 2/2, \;\liminf_{t\to\infty} \sqrt{\log t} R(t,1)=\liminf_{t\to\infty} \sqrt{\log t} \inf_{1\leq s\leq t} R(s,1)=\pi/\sqrt{2},\text{ a.s.}. \]