Anomalous scaling of dynamical large deviations of stationary Gaussian processes

Abstract

Employing the optimal fluctuation method (OFM), we study the large deviation function of longtime averages (1/T ) T /2 −T /2 x n (t)dt, n = 1, 2, . . . , of centered stationary Gaussian processes. These processes are correlated and, in general, non-Markovian. We show that the anomalous scaling with time of the large-deviation function, recently observed for n > 2 for the particular case of the Ornstein-Uhlenbeck process, holds for a whole class of stationary Gaussian processes.

Giant leaps and long excursions: Fluctuation mechanisms in systems with long-range memory

“…This effect might be captured by combining the temporal additivity principle [25] with results for large deviations in Markovian exclusion processes [55], but such an analysis is beyond the scope of the present work. However, we expect the general features to be robust: a large excursion at early times and a rate function scaling as (42). Numerical results confirming the similarity between this non-Markovian SEP and other LIE models are shown in Fig.…”

“…We close this section by noting that (39,42) are both lower bounds on the probability p t (q). Physically, this means that fluctuations can take place by IGL and LIE mechanisms, so fluctuations of a given size q are at least as likely as (39,42) predict.…”

“…Physically, this means that the memory makes large fluctuations less rare [24,25]. See [40][41][42] for some other examples where LDPs with reduced speed are associated with enhanced fluctuations.…”

“…Comparison between the optimally controlled path for the ERW (similar to Fig. 2) and the corresponding generic LIE path used to derive (42). We take a = 0.7 with qt = 0.15 and t = 10 6 .…”

“…as in (42). Since the validity of ( 36) is restricted to small q * , this construction is restricted to small q (strictly positive and fixed as t → ∞).…”