  We study the dynamical phase transitions (DPTs) appearing for a single
Brownian particle without drift. We first explore how first-order DPTs in large
deviations can be found even for a single Brownian particle without any force
upon raising the dimension to higher than four. The DPTs accompany temporal
phase separations in their dynamical paths, which we numerically confirm by
fitting to scaling functions. We next investigate how second-order DPTs can
appear in one-dimensional free Brownian motion by choosing the observable,
which essentially captures the localization transition of the trajectories. We
discuss and confirm that the DPTs predicted for high dimensions can also be
found when considering many Brownian particles at lower dimensions.
