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

Abstract

We study the full distribution of A=∫_{0}^{T}x^{n}(t)dt,n=1,2,⋯, where x(t) is an Ornstein-Uhlenbeck process. We find that for n>2 the long-time (T→∞) scaling form of the distribution is of the anomalous form P(A;T)∼e^{-T^{μ}f_{n}(ΔA/T^{ν})} where ΔA is the difference between A and its mean value, and the anomalous exponents are μ=2/(2n-2) and ν=n/(2n-2). The rate function f_{n}(y), 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 A(t)=∫_{0}^{t}x^{n}(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.