Dynamical large deviations of the fractional Ornstein-Uhlenbeck process

The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation x ˙ ( t ) + γ x = 2 D ξ ( t ) , where ξ ( t ) is the fractional Gaussian noise with the Hurst exponent 0 < H < 1 . For H ≠ 1 / 2 the fOU process is non-Markovian but Gaussian, and it has either vanishing (for H < 1 / 2 ), or divergent (for H > 1 / 2 ) spectral density at zero frequency. For H > 1 / 2 , the fOU is long-correlated. Here we study dynamical large deviations of the fOU process and focus on the area A n = ∫ − T T x n ( t ) d t , n = 1 , 2 , … over a long time window 2T. Employing the optimal fluctuation method, we determine the optimal path of the conditioned process, which dominates the large-A_n tail of the probability distribution of the area, P ( A n , T ) ∼ exp ⁡ [ − S ( A n , T ) ] . We uncover a nontrivial phase diagram of scaling behaviors of the optimal paths and of the action S ( A n ≡ 2 a n T , T ) ∼ T α ( H , n ) a n 2 / n on the (H, n) plane. The phase diagram includes three distinct regions: (i) H > 1 − 1 / n , where α ( H , n ) = 2 − 2 H , and the optimal paths are delocalized in time, (ii) n = 2 and H ⩽ 1 2 , where α ( H , n ) = 1 , and the optimal paths oscillate with an H-dependent frequency, and (iii) H ⩽ 1 − 1 / n and n > 2, where α ( H , n ) = 2 / n , and the optimal paths are strongly localized. We verify our theoretical predictions in large-deviation simulations of the fOU process. By combining the Wang-Landau Monte-Carlo algorithm with the circulant embedding method of generation of stationary Gaussian fields, we were able to measure probability densities as small as 10⁻¹⁷⁰. We also generalize our findings to other stationary Gaussian processes with either divergent, or vanishing spectral density at zero frequency.