  Dynamical phase transitions (DPTs) arise from qualitative changes in the
long-time behavior of stochastic trajectories, often observed in systems with
kinetic constraints or driven out of equilibrium. Here we demonstrate that
first-order DPTs can occur even in the large deviations of a single Brownian
particle without drift, but only when the system's dimensionality exceeds four.
These DPTs are accompanied by temporal phase separations in the trajectories
and exhibit dimension-dependent order due to the threshold behavior for bound
state formation in Schr\"{o}dinger operators. We also discover second-order
DPTs in one-dimensional Brownian motion, characterized by universal exponents
in the rate function of dynamical observables. Our results establish a novel
framework linking classical DPTs to quantum phase transitions.
