A note on the long time asymptotics of the Brownian motion with application to the theory of quantum measurement (Q1376531)

It is proved that for the standard \(d\)-dimensional Brownian motion \(W\) and any \(\beta<3/2- 1/d\), \(\liminf_{t\to\infty} {1\over t^\beta} \left|\int^t_0 W(s)ds\right|= \infty\) almost surely. As an application, the analogous long time asymptotics for the mean coordinate of a quantum particle with continuously observed position is also obtained when \(d> 2\).