Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process

Abstract

We study the full distribution of $A=\int_{0}^{T}x^{n}\left(t\right)dt$, $n=1,2,\dots$, 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 $P\left(A;T\right)\sim e^{-T^{\mu}f_{n}\left(\Delta A/T^{\nu}\right)}$ where $\Delta A$ is the difference between $A$ and its mean value, and the anomalous exponents are $\mu=2/\left(2n-2\right)$, and $\nu=n/\left(2n-2\right)$. The rate function $f_n\left(y\right)$, that 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 $\mathcal{A}(t)=\int_{0}^{t}x^{n}\left(s\right)ds$ and the distribution of $x(t)$ at an intermediate time $t$ conditioned on a given value of $A$. Extensions and implications to other continuous-time systems are discussed.