Anomalous scaling and phase transition in large deviations of dynamical observables of stationary Gaussian processes

Abstract We study large deviations, over a long time window T → ∞ , of the dynamical observables A n = ∫ 0 T x n ( t ) , d t , n = 3 , 4 , … , where x ( t ) is a centered stationary Gaussian process in continuous time. We show that, for short-correlated processes the probability density of A n exhibits an anomalous scaling P ( A n , T ) ∼ exp ⁡ [ − T μ f n ( Δ A n T − ν ) ] at T → ∞ while keeping Δ A n T − ν constant. Here Δ A n is the deviation of A n from its ensemble average. The anomalous exponents µ and ν depend on n and are smaller than 1, whereas the rate function f n ( z ) exhibits a first-order dynamical phase transition (DPT) which resembles condensation transitions observed in many systems. The same type of anomaly and DPT, with the same µ and ν , was previously uncovered for the Ornstein–Uhlenbeck process—the only stationary Gaussian process which is also Markovian. We also uncover an anomalous behavior and a similar DPT in the long-correlated Gaussian processes. However, the anomalous exponents µ and ν are determined in this case not only by n but also by the power-law long-time decay