We study the full distribution of $A={\int }_0^T{x}^n\left(t\right)dt, n=1,2,\cdots$, where $x\left(t\right)$ is an Ornstein-Uhlenbeck process. We find that for $n>2$ the long-time ($T\to \infty$) scaling form of the distribution is of the anomalous form $P(A;T)\sim e^{-T^{\mu }{f}_n(\mathrm{\Delta }A/T^{\nu })}$ where $\mathrm{\Delta }A$ is the difference between $A$ and its mean value, and the anomalous exponents are $\mu =2/(2n-2)$ and $\nu =n/(2n-2)$. The rate function $f_n\left(y\right)$, which we calculate exactly, exhibits a first-order dynamical phase transition which separates between a homogeneous phase that describes the Gaussian distribution of typical fluctuations, and a “condensed” phase that describes the tails of the distribution. We also calculate the most likely realizations of $\mathcal{A}\left(t\right)={\int }_0^t{x}^n\left(s\right)ds$ and the distribution of $x\left(t\right)$ at an intermediate time $t$ conditioned on a given value of $A$. Extensions and implications to other continuous-time systems are discussed.

• Received 18 October 2021
• Accepted 6 January 2022

DOI:https://doi.org/10.1103/PhysRevE.105.014120