Dynamical large deviations of the fractional Ornstein-Uhlenbeck process

Abstract

The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation $\dot{x}(t)+\gamma x=\sqrt{2D}\xi (t)$, where $\xi (t)$ is the fractional Gaussian noise with the Hurst exponent $0<H<1$. For $H\ne 1/2$ 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 $A_n={\int }_{-T}^T{x}^n(t)dt$, $n=1,2,\dots$ 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, $\mathcal{P}(A_n,T)\sim \exp [-S(A_n,T)]$. We uncover a nontrivial phase diagram of scaling behaviors of the optimal paths and of the action $S(A_n\equiv 2a_{n}T,T)\sim T^{\alpha (H,n)}{a}_n^{2/n}$ on the $(H,n)$ plane. The phase diagram includes three regions: (i) $H>1-1/n$, where $\alpha (H,n)=2-2H$, and the optimal paths are delocalized, (ii) $n=2$ and $H\le \frac{1}{2}$, where $\alpha (H,n)=1$, and the optimal paths oscillate with an $H$-dependent frequency, and (iii) $H\le 1-1/n$ and $n>2$, where $\alpha (H,n)=2/n$, 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 diverging, or vanishing spectral density at zero frequency.

21 pages, 11 figures

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Cited by in corpus (3)

• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Beyond the Big Jump: A Perturbative Approach to Stretched-Exponential Processes
• Subleading-order theory for condensation transitions in large deviations of sums of independent and identically distributed random variables