# Distribution of the time of the maximum for stationary processes

Abstract : 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.
Document type :
Journal articles
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https://hal.archives-ouvertes.fr/hal-03389836
Contributor : Claudine Le Vaou Connect in order to contact the contributor
Submitted on : Thursday, October 21, 2021 - 10:44:30 AM
Last modification on : Tuesday, November 16, 2021 - 4:13:07 AM

### Citation

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⟩

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