We study the fluctuation properties of the local time density, ${\rho }_T=\frac{1}{T}{\int }_0^T\delta \left(r\left(t\right)-1\right)dt$, spent by a $d$-dimensional Brownian particle at a spherical shell of unit radius, where $r\left(t\right)$ denotes the radial distance from the particle to the origin. In the large observation time limit, $T\to \infty$, the local time density ${\rho }_T$ obeys the large deviation principle, $P({\rho }_T=\rho )\sim e^{-TI\left(\rho \right)}$, where the rate function $I\left(\rho \right)$ is analytic everywhere for $d\le 4$. In contrast, for $d>4, I\left(\rho \right)$ becomes nonanalytic at a specific point $\rho ={\rho }_c^{\left(d\right)}$, where ${\rho }_c^{\left(d\right)}=d(d-4)/(2d-4)$ depends solely on dimensionality. The singularity signals the occurrence of a first-order dynamical phase transition in dimensions higher than four. Such a transition is accompanied by temporal phase separations in the large deviations of Brownian trajectories. Finally, we validate our theoretical results using a rare-event simulation approach.

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• Brownian motion
• Fluctuations
• Large deviation & rare event statistics