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Distribution of the time of the maximum for stationary processes

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Published 14 October 2021 Copyright © 2021 EPLA
, , Citation Francesco Mori et al 2021 EPL 135 30003 DOI 10.1209/0295-5075/ac19ee

0295-5075/135/3/30003

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.

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10.1209/0295-5075/ac19ee