Time to reach the maximum for a stationary stochastic process – Archive ouverte HAL

Francesco Mori ¹ Satya N. Majumdar ¹ Grégory Schehr ² Satya Majumdar ¹

Francesco Mori, Satya N. Majumdar, Grégory Schehr, Satya Majumdar. Time to reach the maximum for a stationary stochastic process. Physical Review E , 2022, 106 (5), pp.054110. ⟨10.1103/PhysRevE.106.054110⟩. ⟨hal-03903961⟩

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)].

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