标题： 分数阶Ornstein-Uhlenbeck过程的动力学大偏差 显示英文标题

标题： Dynamical large deviations of the fractional Ornstein-Uhlenbeck process

摘要： 分数Ornstein-Uhleneck（fOU）过程由过阻尼Langevin方程$\dot{x}(t)+\gamma x=\sqrt{2 D}\xi(t)$描述，其中$\xi(t)$是具有Hurst指数$0<H<1$的分数高斯噪声。对于$H\neq 1/2$，fOU过程是非马尔可夫但高斯的，并且在零频率处具有消失（对于$H<1/2$）或发散（对于$H>1/2$）的谱密度。对于$H>1/2$，fOU是长相关性的。 此处我们研究fOU过程的动力学大偏差，并关注长时间窗口$2T$上的区域$A_n=\int_{-T}^{T} x^n(t) dt$，$n=1,2,\ldots$。利用最优波动方法，我们确定了条件过程的最优路径，该路径主导了面积的概率分布的大型$A_n$尾部，$\mathcal{P}(A_n,T)\sim \exp[-S(A_n,T)]$。我们揭示了最优路径和作用量$S(A_n\equiv 2 a_n T,T)\sim T^{\alpha(H,n)} a^{2/n}_n$在$(H,n)$平面上的标度行为的非平凡相图。 相图包括三个区域：(i) $H>1-1/n$，其中 $\alpha(H,n)=2-2H$，最优路径是无定域的，(ii) $n=2$ 和 $H\leq \frac{1}{2}$，其中 $\alpha(H,n)=1$，最优路径以与 $H$ 相关的频率振荡，以及 (iii) $H\leq 1-1/n$ 和 $n>2$，其中 $\alpha(H,n)=2/n$，最优路径是强定域的。 我们通过fOU过程的大偏差模拟验证了我们的理论预测。 通过将Wang-Landau蒙特卡洛算法与平稳高斯场生成的循环嵌入方法相结合，我们能够测量小至$10^{-170}$的概率密度。 我们还将这些发现推广到其他在零频率处具有发散或消失谱密度的平稳高斯过程。 显示英文摘要

摘要： The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation $\dot{x}(t)+\gamma x=\sqrt{2 D}\xi(t)$, where $\xi(t)$ is the fractional Gaussian noise with the Hurst exponent $0<H<1$. For $H\neq 1/2$ 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 $A_n=\int_{-T}^{T} x^n(t) dt$, $n=1,2,\ldots$ 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 $S(A_n\equiv 2 a_n T,T)\sim T^{\alpha(H,n)} a^{2/n}_n$ 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 $H\leq \frac{1}{2}$, where $\alpha(H,n)=1$, and the optimal paths oscillate with an $H$-dependent frequency, and (iii) $H\leq 1-1/n$ 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.