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

^(a) francesco.mori@universite-paris-saclay.fr (corresponding author)

Received: 27 May 2021
Accepted: 2 August 2021

Abstract

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.