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

We study the full distribution of A = LT xn(t)dt, n = 1, 2, ... , where x(t) is an Ornstein-Uhlenbeck process. We find that for n > 2 the long-time (T -> infinity) scaling form of the distribution is of the anomalous form P(A; T ) similar to e-T mu fn(A/T nu ) where 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 fn(y), which we calculate exactly, exhibits a firstorder 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 A(t) = j 0 txn(s)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.

Automated summary

In this study, we investigate the distribution of A = LT xn(t)dt, where x(t) is an Ornstein-Uhlenbeck process. We find that for n > 2, the long-time scaling form of the distribution exhibits an anomalous behavior. By calculating the exact rate function, we identify a first-order dynamical phase transition that separates the Gaussian distribution of typical fluctuations from the condensed phase describing the tails of the distribution.