Dynamical large deviations of the fractional Ornstein-Uhlenbeck process
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2024
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| author | Valov, Alexander Meerson, Baruch |
| author_facet | Valov, Alexander Meerson, Baruch |
| contents | The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation $\dot{x}(t)+γx=\sqrt{2 D}ξ(t)$, where $ξ(t)$ 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 $S(A_n\equiv 2 a_n T,T)\sim T^{α(H,n)} a^{2/n}_n$ on the $(H,n)$ plane. The phase diagram includes three regions: (i) $H>1-1/n$, where $α(H,n)=2-2H$, and the optimal paths are delocalized, (ii) $n=2$ and $H\leq \frac{1}{2}$, where $α(H,n)=1$, and the optimal paths oscillate with an $H$-dependent frequency, and (iii) $H\leq 1-1/n$ 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 diverging, or vanishing spectral density at zero frequency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02398 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamical large deviations of the fractional Ornstein-Uhlenbeck process Valov, Alexander Meerson, Baruch Statistical Mechanics The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation $\dot{x}(t)+γx=\sqrt{2 D}ξ(t)$, where $ξ(t)$ 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 $S(A_n\equiv 2 a_n T,T)\sim T^{α(H,n)} a^{2/n}_n$ on the $(H,n)$ plane. The phase diagram includes three regions: (i) $H>1-1/n$, where $α(H,n)=2-2H$, and the optimal paths are delocalized, (ii) $n=2$ and $H\leq \frac{1}{2}$, where $α(H,n)=1$, and the optimal paths oscillate with an $H$-dependent frequency, and (iii) $H\leq 1-1/n$ 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 diverging, or vanishing spectral density at zero frequency. |
| title | Dynamical large deviations of the fractional Ornstein-Uhlenbeck process |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2412.02398 |