论文标题
是时候达到固定随机过程的最大值
Time to reach the maximum for a stationary stochastic process
论文作者
论文摘要
我们考虑固定持续时间$ t $的一维固定时间序列。我们调查了该过程在时间间隔$ [0，t] $之内达到全局最大值的时间$ t _ {\ rm m} $。通过使用路径分解技术，我们计算$ t _ {\ rm m} | t）的概率密度函数$ p（t _ {\ rm m} | t）$ t _ {\ rm m} $的多个过程，它们要么处于平衡状态（例如Ornstein-uhlenbeck过程），要么是equilibibium（例如，与brownian resovers resostic resets snking snking snking snking snking snking snking snking snking snking snk s n equilibibrium equililibium（我们表明，对于平衡过程，$ p（t _ {\ rm m} | t）$的分布始终是在中点周围$ t _ {\ rm m} = t/2 $的对称性，这是时间逆向对称的结果。该特性可用于检测固定时间序列中的非平衡波动。此外，对于限制电势中的扩散粒子，我们表明缩放分布$ p（t _ {\ rm m} | t）$变为通用，即在后期独立于电势的细节。此分布$ p（t _ {\ rm m} | t）$在“ bulk” $ 1 \ ll t _ {\ rm m} \ ll t $中变得均匀，并且在“边缘制度”中具有非平凡的通用形状。这些结果中的一些已在最近的一封信[Europhys。 Lett。 {\ bf 135}，30003（2021）]。
We consider a one-dimensional stationary time series of fixed duration $T$. We investigate the time $t_{\rm m}$ at which the process reaches the global maximum within the time interval $[0,T]$. By using a path-decomposition technique, we compute the probability density function $P(t_{\rm m}|T)$ of $t_{\rm m}$ for several processes, that are either at equilibrium (such as the Ornstein-Uhlenbeck process) or out of equilibrium (such as Brownian motion with stochastic resetting). We show that for equilibrium processes the distribution of $P(t_{\rm m}|T)$ is always symmetric around the midpoint $t_{\rm m}=T/2$, as a consequence of the time-reversal symmetry. This property can be used to detect nonequilibrium fluctuations in stationary time series. Moreover, for a diffusive particle in a confining potential, we show that the scaled distribution $P(t_{\rm m}|T)$ becomes universal, i.e., independent of the details of the potential, at late times. This distribution $P(t_{\rm m}|T)$ becomes uniform in the "bulk" $1\ll t_{\rm m}\ll T$ and has a nontrivial universal shape in the "edge regimes" $t_{\rm m}\to0$ and $t_{\rm m} \to T$. Some of these results have been announced in a recent Letter [Europhys. Lett. {\bf 135}, 30003 (2021)].
