Volume 135, Number 3, August 2021
|Number of page(s)||7|
|Published online||14 October 2021|
Distribution of the time of the maximum for stationary processes
1 LPTMS, CNRS, Univ. Paris- Sud, Université Paris-Saclay - 91405 Orsay, France
2 Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS, UMR 7589 4 Place Jussieu, 75252 Paris Cedex 05, France
Received: 27 May 2021
Accepted: 2 August 2021
We consider a one-dimensional stationary stochastic process of duration T. We study the probability density function (PDF) of the time at which reaches its global maximum. By using a path integral method, we compute 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 , 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 and has a nontrivial edge scaling behavior for (and when ), that we compute exactly. Moreover, we show that for any equilibrium process the PDF is symmetric around , i.e., . This symmetry provides a simple method to decide whether a given stationary time series is at equilibrium or not.
© 2021 EPLA
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.