Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $\mathrm{\mid }x\mathrm{\mid }$ potential

We study the fluctuations of the area $A={\int }_0^{T}x(t)dt$ under a one-dimensional Brownian motion $x(t)$ in a trapping potential $\sim \mathrm{\mid }x\mathrm{\mid }$, at long times $T\to \mathrm{\infty }$. We find that typical fluctuations of $A$ follow a Gaussian distribution with a variance that grows linearly in time (at large $T$), as do all higher cumulants of the distribution. However, large deviations of $A$ are not described by the ``usual’’ scaling (i.e., the large deviations principle), and are instead described by two different anomalous scaling behaviors: Moderately-large deviations of $A$, obey the anomalous scaling $P(A;T)\sim e^{-T^{1\mathrm{/}3}f\left(A\mathrm{/}{T}^{2\mathrm{/}3}\right)}$ while very large deviations behave as $P(A;T)\sim e^{-T\mathrm{\Psi }\left(A\mathrm{/}{T}^2\right)}$. We find the associated rate functions $f$ and $\mathrm{\Psi }$ exactly. Each of the two functions contains a singularity, which we interpret as dynamical phase transitions of the first and third order, respectively. We uncover the origin of these striking behaviors by characterizing the most likely scenario(s) for the system to reach a given atypical value of $A$. We extend our analysis by studying the absolute area $B={\int }_0^T\mathrm{\mid }x(t)\mathrm{\mid }dt$ and also by generalizing to higher spatial dimension, focusing on the particular case of three dimensions.