Distribution of the time of the maximum for stationary processes – 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. Distribution of the time of the maximum for stationary processes. EPL – Europhysics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2021, 135 (3), pp.30003. ⟨10.1209/0295-5075/ac19ee⟩. ⟨hal-03389836⟩

We consider a one-dimensional stationary stochastic process $x(\tau)$ of duration $T$. We study the probability density function (PDF) $P(t_{\rm m}|T)$ of the time $t_{\rm m}$ at which $x(\tau)$ reaches its global maximum. By using a path integral method, we compute $P(t_{\rm m}|T)$ for a number of equilibrium and nonequilibrium stationary processes, including the Ornstein-Uhlenbeck process, Brownian motion with stochastic resetting and a single confined run-and-tumble particle. For a large class of equilibrium stationary processes that correspond to diffusion in a confining potential, we show that the scaled distribution $P(t_{\rm m}|T)$, for large $T$, has a universal form (independent of the details of the potential). This universal distribution is uniform in the « bulk », i.e., for $0 \ll t_{\rm m} \ll T$ and has a nontrivial edge scaling behavior for $t_{\rm m} \to 0$ (and when $t_{\rm m} \to T$), that we compute exactly. Moreover, we show that for any equilibrium process the PDF $P(t_{\rm m}|T)$ is symmetric around $t_{\rm m}=T/2$, i.e., $P(t_{\rm m}|T)=P(T-t_{\rm m}|T)$. This symmetry provides a simple method to decide whether a given stationary time series $x(\tau)$ is at equilibrium or not.

  • 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. LPTHE – Laboratoire de Physique Théorique et Hautes Energies

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